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Geometric Functional Analysis and Probability Seminar

ThursdayJul 13, 202313:30
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:Maximilian FelsTitle:The phase diagram of the CREM at complex temperatureAbstract:opens in new windowin html    pdfopens in new window

The continuous random energy model (CREM) was introduced by Bovier and Kurkova (2004) as a generalization of Derrida's generalized random energy model (GREM) (1985). Both are (mathematically tractable) spin glass models in which multiple freezing transitions can be observed. CREM defined on a Galton-Watson tree coincides with variable-speed branching Brownian motion (BBM) defined on the identical tree. In this talk, I will introduce the latter model and present recent and ongoing work with Lisa Hartung and Anton Klimovsky concerning the log-partition function of the CREM on a Galton-Watson tree at complex temperature. In particular, one obstacle of working in the complex plane is the loss of Gaussian comparison tools which most of the previous analysis of variable-speed BBM relied on. I will present a novel coupling which allows us to overcome this technical issue which is of its own interest.

ThursdayJul 06, 202313:30
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:Gidi AmirTitle:The firefighter problem, Cayley graphs and the branching number of intermediate growth treesAbstract:opens in new windowin html    pdfopens in new window

In the fire-fighter model, a fire spreads on the vertices of a graph starting from some initial fire. Each turn all ``non-protected" vertices neighbouring the fire catch fire and burn forever. The ``player" can protect f(n) vertices on his n'th turn and these vertices never catch fire. Given a graph G, The ``asymptotic" fire-fighter problem asks for which $f(n)$ can the fire be contained in a finite set for any initial fire.

This is a quasi-isometry invariant of G, and is especially interesting for Cayley graphs.

 

We will first discuss the problem on Polynomial growth groups, where in joint work with Gady Kozma and Rangel Baldasso we show that for groups of growth n^d the threshold is ~n^{d-2}, answering a conjecture of Devline and Hartke.

We will then move to discuss the case of exponential growth groups where F. Lehner proved that the growth rate is also the order of the threshold for fire-fighting.

This leaves open the case of intermediate growth groups. In joint work with Shangjie Yang we get the correct threshold for a family of intermediate growth groups. To do so we introduce a notion of branching numbers for intermediate growth trees (IBN), which acts as the threshold for several random processes on such trees. We relate the IBN it to firefighting and find and analyze a good tree inside these groups.

 

If time permits we will also briefly discuss some other fire-fighting related problems.

 

Based on joint works with R. Baldasso, G. Kozma, M. Gerasimova and S. Yang.

ThursdayJun 29, 202313:30
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:Oren YakirTitle:Fluctuations in the logarithmic energy for zeros of random polynomials on the sphereAbstract:opens in new windowin html    pdfopens in new window
Smale's 7th Problem asks for an efficient algorithm to generate a configuration of n points on the sphere that nearly minimizes the logarithmic energy. As a candidate starting configuration for this problem, Armentano, Beltran and Shub considered the roots of the random elliptic polynomial of degree n and computed the expected logarithmic energy. We study fluctuations of the logarithmic energy of this random configuration and prove a central limit theorem. Our analysis shows that the energy is well concentrated around its mean on the scale of \sqrt(n). The talk is based on a joint work with Marcus Michelen.
ThursdayJun 22, 202313:30
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:Jonathan HermonTitle:Universality of cutoff for graphs with a random matchingAbstract:opens in new windowin html    pdfopens in new window

(Joint work with Perla Sousi and Allan Sly.) A sequence of Markov chains is said to exhibit cutoff if it exhibits an abrupt convergence to equilibrium. Loosely speaking, this means that the epsilon mixing time (i.e. the first time the worst-case total-variation distance from equilibrium is at most epsilon) is asymptotically independent of epsilon. An emerging paradigm is that cutoff is related to entropic concentration. In the case of random walks on random graphs G=(V,E), the entropic time can be defined as the time at which the entropy of random walk on some auxiliary graph (often the Benjamini-Schramm limit) is log |V|. Previous works established cutoff at the entropic time in the case of the configuration model. We consider a random graph model in which a random graph G'=(V,E') is obtained from a given graph G=(V,E) by picking a random perfect matching of the vertices, and adding an edge between each pair of matched vertices. We prove cutoff at the entropic time, provided G is of bounded degree and its connected components are of size at least 3. Previous works were restricted to the case that the random graph is locally tree-like.

We also consider the case that the added edges from the perfect matching have a weight epsilon < 1 which may depend on |V| and in many cases we determine for which range of values of epsilon there is cutoff.

ThursdayJun 08, 202313:30
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:Asaf NachmiasTitle:Scaling limit of high-dimensional uniform spanning treesAbstract:opens in new windowin html    pdfopens in new window

Szekeres proved in 1982 that the diameter (length of longest path) of a uniformly drawn labeled tree on n vertices normalized by the square root of n converges in distribution to an explicitly described distribution. This random tree is just a uniformly chosen spanning tree of the complete graph on n vertices. What if one changes the underlying graph from the complete graph to, say, the hypercube {0,1}^n, or an expander graph, or in cubic lattices of fixed but high dimensions? Our result shows that one gets the same limiting distribution of the diameter. In fact much more is true: any reasonable "global" property of these random trees will have the same limiting distribution as a uniformly chosen labelled tree, moreover, these distributions can be explicitly described via Aldous' 1991 continuum random tree.

Joint work with Eleanor Archer and Matan Shalev.

ThursdayJun 01, 202313:30
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:Eviatar ProcacciaTitle:Fluctuations of Stationary Hastings Levitov are log-correlatedAbstract:opens in new windowin html    pdfopens in new window

Stationary Hastings Levitov (SHL) provides a good model for diffusion limited aggregation growing on a long fiber. Due to regularity properties of the conformal map attaching a slit to the upper half plane, no particle normalization is required to avoid particle size blowup, such as one obtains for the Hastings Levitov process growing on a disk. In this talk we will discuss a work in progress with Noam Berger (TUM) showing that the fluctuations of SHL around it's deterministic growth rate is log-correlated for points of distance smaller than the time of the process and uncorrelated for points of distance larger than the time. The local maxima of this field has a physical interpretation as it corresponds to the longest arms in the aggregate. Due to the tractability of the process, one can also bound the infinitesimal generator and in turn the exponential moments.  

MondayMay 22, 202313:00
Geometric Functional Analysis and Probability SeminarRoom 1
Speaker:Yuri LimaTitle:Hypergraph-based Polya urnsAbstract:opens in new windowin html    pdfopens in new windowPlease Note The Unusual Room And Day!

A paper of Benaim, Benjamini, Chen and Lima introduced a model of Polya urns with graph-based interactions. For these systems, the proportions of balls in the bins converge almost surely. In this talk, I will discuss the hypergraph-based version of this model, and report on partial results about the convergence of the proportions of balls in the bins, in terms of the incidence matrix of the hypergraph.

ThursdayMay 18, 202313:30
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:Alexander GrigoryanTitle:Heat kernels of diffusions and jump processes on fractalsAbstract:opens in new windowin html    pdfopens in new window
We discuss elements of Analysis on Ahlfors-regular metric spaces, in particular, on fractals, based on the notion of the heat kernel. Such spaces are characterized by two parameters: the Hausdorff dimension and the walk dimension, where the latter determines the space/time scaling for a diffusion process. We present various approaches to the notion of the walk dimension, including those via Besov function spaces and via Markov jump processes. We also discuss heat kernel bounds for diffusion and jump processes.
ThursdayMay 04, 202313:30
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:Shmuel FriedlandTitle:The Saga of the matrix Grothendieck inequalityAbstract:opens in new windowin html    pdfopens in new window

ThursdayApr 27, 202313:30
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:Michael HofstetterTitle:Extreme values of non-Gaussian fieldsAbstract:opens in new windowin html    pdfopens in new windowThe seminar has been rescheduled for next week at 13:30 (27.4.23)

In recent years there has been significant progress in the study of extreme values of log-correlated

Gaussian fields, thanks to the work of Bramson, Ding, Roy, Zeitouni and Biskup, Louidor. For instance, it has been shown that for the discrete Gaussian free field (DGFF) in d=2 and for log-correlated Gaussian fields the limiting law of the centred maximum is a randomly shifted Gumbel distribution.

 

In this talk I will present analogous results for non-Gaussian fields such as the sine-Gordon field and the \Phi^4 field in d = 2. The main tool is a coupling at all scales between the field of interest and the DGFF which emerges from the Polchinski renormalisation group approach as well as the Boue-Dupuis variational formula. The talk is based on joint works with Roland Bauerschmidt and Trishen Gunaratnam, Nikolay Barashkov.

ThursdayFeb 09, 202313:30
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:Leonid MytnikTitle:On the coming down from infinity of coalescing Brownian motionsAbstract:opens in new windowin html    pdfopens in new window

Consider a system of Brownian particles on the real line where each pair of particles coalesces at a certain rate according to their intersection local time.

Assume that initially there are infinitely many particles in the system. We give conditions for the number of particles to come down from infinity.

We also identify the rate of the coming down from infinity for different initial configurations.

This is a joint work with C. Barnes and Z. Sun.

ThursdayFeb 02, 202313:30
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:Naomi FeldheimTitle:A phase transition in zero count of stationary Gaussian processesAbstract:opens in new windowin html    pdfopens in new window
Let f be a stationary Gaussian process on R with compactly supported spectral measure. The expected number of zeroes is computed by the celebrated Kac-Rice formula, and much is known about their typical behavior. In this talk we are interested in their large deviations: What is the probability to see many more zeroes (overcrowding) or many less zeroes (undercrowding) than expected in a long interval? We show that overcrowding and undercrowding probabilities exhibit a phase transition between exponential and Gaussian behavior. The critical points of the transition are determined by the support of the spectrum. This result complements previous bounds given by Basu-Dembo-F.-Zeitouni and by Priya. Based on ongoing joint work with Ohad Feldheim and Lakshmi Priya.
ThursdayJan 26, 202313:30
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:Barak WeissTitle:New bounds on lattice covering volumes, and nearly uniform coversAbstract:opens in new windowin html    pdfopens in new window
Let L be a lattice in R^n and let K be a convex body. The covering volume of L with respect to K is the minimal volume of a dilate rK, such that L+rK = R^n, normalized by the covolume of L. Pairs (L,K) with small covering volume correspond to efficient coverings of space by translates of K, where the translates lie in a lattice. Finding upper bounds on the covering volume as the dimension n grows is a well studied problem, with connections to practical questions arising in computer science and electrical engineering. In a recent paper with Or Ordentlich (EE, Hebrew University) and Oded Regev (CS, NYU) we obtain substantial improvements to bounds of Rogers from the 1950s. In another recent paper, we obtain bounds on the minimal volume of nearly uniform covers (to be defined in the talk). The key to these results are recent breakthroughs by Dvir and others regarding the discrete Kakeya problem. I will give an overview of the questions and results. No mathematical prerequisites will be assumed (beyond the concept of Lebesgue measure and vector spaces over finite fields).
ThursdayJan 19, 202313:30
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:Noam LifshitzTitle:Product free sets in groupsAbstract:opens in new windowin html    pdfopens in new window
A subset $A$ of a group is said to be product free if for all elements $a,b$ in $A$ their product $ab$ is not in $A$. In 1985 Babai and Sos posed the problem of determining the largest product free sets in the alternating group. We solve this problem completely when $n$ is sufficiently large. Our proof combines representation theory with a recent machinery we developed called 'hypercontractvity for global functions'.
ThursdayJan 12, 202313:30
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:Aleksei KulikovTitle:Norms of embeddings between spaces of log-subharmonic functions on surfaces of constant curvatureAbstract:opens in new windowin html    pdfopens in new window

ThursdayJan 05, 202313:30
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:Jonathan HermonTitle:On the universality of fluctuations for the cover timeAbstract:opens in new windowin html    pdfopens in new window
What is the structure of the set of the last few points visited by a random walk on a graph? We show that on vertex-transitive graphs of bounded degree, this set is decorrelated (it is close to a product measure in total variation) if and only if a simple geometric condition on the diameter of the graph holds. In this case, the cover time has universal fluctuations: properly scaled, this time converges to a Gumbel distribution. To prove this result we rely on recent progress in geometric group theory (about quantitative versions of Gromov's Theorem for finite vertex-transitive graphs), and we prove refined quantitative estimates showing that the hitting time of a small set of vertices is typically approximately an exponential random variable. This talk is based on joint work with Nathanael Berestycki and Lucas Teyssier.
ThursdayDec 29, 202214:30
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:Dan MikulincerTitle:Lipschitz mass transportAbstract:opens in new windowin html    pdfopens in new window
A central question in the field of optimal transport studies optimization problems involving two measures on a common metric space, a source and a target. The goal is to find a mapping from the source to the target, in a way that minimizes distances. A remarkable fact discovered by Caffarelli is that, in some specific cases of interest, the optimal transport maps on a Euclidean metric space are Lipschitz. Lipschitz regularity is a desirable property because it allows for the transfer of analytic properties between measures. This perspective has proven to be widely influential, with applications extending beyond the field of optimal transport. In this talk, we will further explore the Lipschitz properties of transport maps. Our main observation is that, when one seeks Lipschitz mappings, the optimality conditions mentioned above do not play a major role. Instead of minimizing distances, we will consider a general construction of transport maps based on interpolation of measures, and introduce a set of techniques to analyze the Lipschitz constant of this construction. In particular, we will go beyond the Euclidean setting and consider Riemannian manifolds as well as infinite-dimensional spaces. Some applications, such as functional inequalities and normal approximations will also be discussed.
ThursdayDec 29, 202213:30
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:Rachel GreenfeldTitle:The structure of translational tilingsAbstract:opens in new windowin html    pdfopens in new window
Translational tiling is a covering of a space (e.g., Euclidean space) using translated copies of a building block, called a "tile'', without any positive measure overlaps. What are the possible ways that a space can be tiled? One of the most well known conjectures in this area is the periodic tiling conjecture. It asserts that any tile of Euclidean space can tile the space periodically. This conjecture was posed 35 years ago and has been intensively studied over the years. In a joint work with Terence Tao, we disprove the periodic tiling conjecture in high dimensions. In the talk, I will motivate this result and discuss our proof.
ThursdayDec 15, 202213:30
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:Yeor HafoutaTitle:A Berry-Esseen theorem in L^p under weak dependenceAbstract:opens in new windowin html    pdfopens in new window
The classical Berry-Esseen theorem provides optimal rates in the central limit theorem (CLT) for partial sums of iid random variables. Since then there have been many extensions to ``weakly dependent" (aka mixing) random variables. A related question is the accuracy of approximation by Guassians in L^p (after coupling). The validity of such optimal L^p rates was an open problem by E. Rio, which was recently (2018) solved by S. Bobkov for independent random variables. In the talk I will present recent results concerning optimal CLT rates in L^p for a variety of weakly dependent random variables like (inhomogeneous) Markov chains, products of random matrices and partially hyperbolic dynamical systems. We will also discuss relations with almost sure rates of approximation by Gaussian random variables (i.e. rates in the almost sure invariance principle). Edgeworth expansions provide better than optimal CLT rates, with appropriate correction terms. Our proofs require non-uniform versions of such expansions for weakly dependent random variables, which are of independent interest and have several other applications. In particular, we obtain a non-uniform Berry-Esseen theorem for weakly dependent random variables.
ThursdayJul 14, 202213:30
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:Peter VeermanTitle:Rank Driven DynamicsAbstract:opens in new windowin html    pdfopens in new window
We investigate a class of models related to the Bak-Sneppen (BS) model, initially proposed to study evolution. The BS model is extremely simple and yet captures some forms of complex behavior such as self-organized criticality that is often observed in physical and biological systems. In this model, random fitnesses in [0, 1] are associated to N agents located at the vertices of a graph G, in our case a cycle. Their fitnesses are ranked from worst (0) to best (1). At every time-step the agent with the lowest fitness and its neighbors on the graph G are replaced by new agents with random fitnesses. We approximate the dynamics by making a simplifying independence assumption. We use order statistics to define a dynamical system on the set of cumulative distribution functions R : [0, 1] -> [0, 1] that mimics the evolution of the distribution of the fitnesses in these rank-driven models, among which the BS model. We then show that this dynamical system reduces to a 1-dimensional polynomial map. Using an additional conjecture we can then find the limiting distribution as a function of the initial conditions. Roughly speaking, this ansatz says that the bulk of the replacements in the Bak-Sneppen model occur in a decreasing fraction of the population as the number N of agents tends to infinity. Agreement with experimental results of the BS model is excellent.
ThursdayJul 07, 202213:30
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:Oren LouidorTitle:TBDAbstract:opens in new windowin html    pdfopens in new window
ThursdayJul 07, 202213:30
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:Oren LouidorTitle:A limit law for the most favorite point of a simple random walk on a regular treeAbstract:opens in new windowin html    pdfopens in new window
We consider a continuous-time random walk on a regular tree of finite depth and study its favorite points among the leaf vertices. We prove that, for the walk started from a leaf vertex and stopped upon hitting the root, as the depth of the tree tends to infinity the maximal time spent at any leaf converges, under suitable scaling and centering, to a randomly-shifted Gumbel law. The random shift is characterized using a derivative-martingale like object associated with square-root local-time process on the tree. Joint work with Marek Biskup (UCLA).
ThursdayJun 30, 202213:30
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:Uzy SmilanskyTitle:Can one hear a real symmetric matrix? Abstract:opens in new windowin html    pdfopens in new window
The question asked in the title is addressed from two points of view: First, we show that providing enough (term to be explained) spectral data, suffices to reconstruct uniquely generic (term to be explained) matrices. The method is well defined but requires somewhat cumbersome computations. Second, restricting the attention to banded matrices with band-width much smaller than the dimension, one can provide more spectral data than the number of unknown matrix elements. We make use of this redundancy to reconstruct generic banded matrices in a much more straight- forward fashion where the “cumbersome computations” become unnecessary. Explicit criteria for a matrix to be in the non-generic set are provided.
ThursdayJun 23, 202213:30
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:Matan EilatTitle:Rigidity of Riemannian embeddings of discrete metric spacesAbstract:opens in new windowin html    pdfopens in new window
Let M be a complete, connected Riemannian surface and suppose that S is a discrete subset of M. What can we learn about M from the knowledge of all distances in the surface between pairs of points of S? We prove that if the distances in S correspond to the distances in a 2-dimensional lattice, or more generally in an arbitrary net in R^2, then M is isometric to the Euclidean plane. We thus find that Riemannian embeddings of certain discrete metric spaces are rather rigid. A corollary is that a subset of Z^3 that strictly contains a two-dimensional lattice cannot be isometrically embedded in any complete Riemannian surface. This is a joint work with B. Klartag.
ThursdayJun 09, 202213:30
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:Maksim ZhukovskiiTitle:Limit theorems for locally stable tree parametersAbstract:opens in new windowin html    pdfopens in new window
For a uniform random labelled tree, we find the limiting distribution of tree parameters which are stable (in some sense) with respect to local perturbations of the tree. The proof is based on the martingale central limit theorem and the Aldous Broder algorithm. In particular, our general result implies the asymptotic normality of the number of occurrences of any given small pattern and the asymptotic log-normality of the number of automorphisms.
ThursdayJun 02, 202213:30
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:Itay LondnerTitle:Tiling the integers with translates of one tile: the Coven-Meyerowitz tiling conditionsAbstract:opens in new windowin html    pdfopens in new window
It is well known that if a finite set of integers A tiles the integers by translations, then the translation set must be periodic, so that the tiling is equivalent to a factorization A+B=Z_M of a finite cyclic group. Coven and Meyerowitz (1998) proved that when the tiling period M has at most two distinct prime factors, each of the sets A and B can be replaced by a highly ordered "standard" tiling complement. It is not known whether this behaviour persists for all tilings with no restrictions on the number of prime factors of M. In joint work with Izabella Laba (UBC), we proved that this is true for all sets tiling the integers with period M=(pqr)^2. In my talk I will discuss this problem and introduce some ideas from the proof.
ThursdayMay 26, 202213:30
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:Dor ElboimTitle:Long lines in subsets of large measure in high dimensions.Abstract:opens in new windowin html    pdfopens in new window

ThursdayMay 19, 202213:30
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:David EllisTitle:Product mixing on compact Lie groupsAbstract:opens in new windowin html    pdfopens in new window

We obtain almost-sharp product mixing inequalities for certain infinite families of compact Lie groups: viz., the special orthogonal groups, the special unitary groups, the spin groups and the compact symplectic groups. Our proofs rely on Fourier analysis and hypercontractive inequalities on these groups, which can be proven by two rather different methods: either an approach based on the Bakry-Emery criterion and Ricci curvature bounds, or a probabilistic coupling approach. We also employ combinatorial tools (for example to analyse Lie group representations). As an application we make progress on a question of Gowers concerning the largest product-free subsets of the special unitary groups SU(n). Our results also have applications in physics, for example in the analysis of Kac's random walk on the special orthogonal group, which is a toy model for Boltzmann gas dynamics.

 

Based on joint work with Guy Kindler (HUJI), Noam Lifshitz (HUJI) and Dor Minzer (MIT).

 

ThursdayMay 12, 202213:30
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:Mark RudelsonTitle:When a system of real quadratic equations has a solutionAbstract:opens in new windowin html    pdfopens in new window
The existence and the number of solutions of a system of polynomial equations in $n$ variables over an algebraically closed field is a classical topic in algebraic geometry. Much less is known about the existence of solutions of a system of polynomial equations over reals. Any such problem can be reduced to a system of quadratic equations by introducing auxiliary variables. Due to the generality of the problem, a computationally efficient algorithm for determining whether a real solution of a system of quadratic equations exists is believed to be impossible. We will discuss a simple and efficient sufficient condition for the existence of a solution. While the problem and the condition are of algebraic nature, the proof relies on Fourier analysis and concentration of measure. Joint work with Alexander Barvinok.
ThursdayApr 07, 202213:30
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:Oren YakirTitle:Random Weierstrass Zeta-functionsAbstract:opens in new windowin html    pdfopens in new window

ThursdayDec 23, 202113:30
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:Jacob ShapiroTitle:Depinning in the integer-valued Gaussian field and the BKT phase of the 2D Villain modelAbstract:opens in new windowin html    pdfopens in new windowzoom link as well: https://weizmann.zoom.us/j/96250855312?pwd=empqTUZGcU5TcHY5V1FsL2xDVnZwZz09
It is shown that the Villain model of two-component spins over two dimensional lattices exhibits slow, non-summable, decay of correlations at any temperature at which the dual integer-valued Gaussian field exhibits depinning. For the latter, we extend the recent proof by Lammers of the existence of a depinning transition in the integer-valued Gaussian field in two-dimensional cubic graphs to all doubly periodic graphs, in particular to Z^2. Taken together these two statements yield a new perspective on the Berezinskii Kosterlitz Thouless phase transition in the Villain model, and complete a new proof of depinning in two-dimensional integer-valued height functions. Based on joint work with: Michael Aizenman, Matan Harel and Ron Peled.
ThursdayDec 16, 202113:30
Geometric Functional Analysis and Probability Seminar
Speaker:Sarai Hernandez-TorresTitle:The chemical distance of random interlacements in the low intensity regime.Abstract:opens in new windowin html    pdfopens in new window

Random interlacements (RI) is a Poissonian soup of doubly-infinite random walk trajectories on Z^d. A parameter u > 0 controls the intensity of the Poisson point process. In a natural way, the model defines a long-range percolation on the edges of Z^d. We thus obtain the random interlacements graph, composed of those edges traversed by a trajectory in RI. This talk focuses on the chemical distance of the random interlacements graph in dimensions d \geq 5. In this setting, I will present a proof of novel upper and lower asymptotic bounds on the chemical distance for u << 1. This is a joint work with E. Procaccia and R. Rosenthal.

ThursdayDec 09, 202113:30
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:Uri GrupelTitle:Combinatorial diameter of random polytopes.Abstract:opens in new windowin html    pdfopens in new window
Given a polytope, we define the combinatorial diameter as the maximal shortest path (with respect to the number of edges we traverse) between any two vertices. Bounding the combinatorial diameter was motivated by the study of the simplex method. The polynomial Hirsch conjecture states that the combinatorial diameter of any polytope in dimension n with m facets should be bounded by a polynomial of n and m. We will discuss a class of random polytopes, generated by a Poisson point process on the sphere. We will see asymptotic bounds for the diameter, and compare it to other known bounds. Based on joint work with G. Bonnet, D. Dadush, S. Huiberts and G. Livshyts.
ThursdayDec 02, 202113:30
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:Clément CoscoTitle:High moments of partition function for 2D polymers in the weak disorder regime (joint with Ofer Zeitouni)Abstract:opens in new windowin html    pdfopens in new window
We consider the model of directed polymers in dimension 1+2, where we take the temperature of the model to infinity with the volume, in such a way that the partition function stays bounded in L2. It is well known that the diffusively rescaled log-partition function converges to a Gaussian log-correlated field. One natural question is to understand the behavior of the maximum of this rescaled field, which is related to the problem of understanding the probability distribution of the favorite point of the polymer trajectory. One major issue is that the field itself (before taking the limit) is not Gaussian, and one has to quantify how close it is to a Gaussian field. A direction towards this is to compute moments of the rescaled logarithm field, which amounts to taking large moments (going to infinity with the volume) of the (non-log) partition function. I will explain how we are able to show that the moments coincide with Gaussian moments up to some threshold, and discuss the background and open questions.
ThursdayNov 25, 202113:30
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:Florian SchweigerTitle:Extrema of log-correlated Gaussian fields in random environmentAbstract:opens in new windowin html    pdfopens in new window

Log-correlated Gaussian fields are a class of random interface models which show a very rich behaviour. In the last decade there has been great progress in the understanding of the extrema of these fields, and it has been shown for various examples that the recentred maximum of the field converges in distribution.
In this talk I will briefly summarize these results, and then explain how to extend them to some fields in an environment which is itself random. In particular, I will discuss the example of the Gaussian free field on the two-dimensional supercritical percolation cluster. This requires sharp quenched estimates on the Green’s function of the Laplacian on the cluster.

ThursdayNov 18, 202113:30
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:Yotam SmilanskyTitle:Order and disorder in multiscale substitution tilingsAbstract:opens in new windowin html    pdfopens in new windowThe lecture will be projected in zoom: https://weizmann.zoom.us/j/94740221234?pwd=aHJ2NlM4SGprSHVWVUR4L2RBUGpndz09
The study of aperiodic order and mathematical models of quasicrystals is concerned with ways in which disordered structures can nevertheless manifest aspects of order. In the talk I will describe examples such as the aperiodic Penrose and pinwheel tilings, together with several geometric, functional, dynamical and spectral properties that enable us to measure how far such constructions are from demonstrating lattice-like behavior. A particular focus will be given to new results on multiscale substitution tilings, a class of tilings that was recently introduced jointly with Yaar Solomon.
ThursdayAug 19, 202111:00
Geometric Functional Analysis and Probability Seminar
Speaker:Sasha SodinTitle:Lower bounds on the eigenfunctions of random Schroedinger operators in a stripAbstract:opens in new windowin html    pdfopens in new windowZoom Meeting: https://weizmann.zoom.us/j/5117633893?pwd=TjZHajlNQ1RScldPbEZ4Ny80TEUyQT09
It is known that the eigenfunctions of a random Schroedinger operator in a strip (the direct product of the integer line and a finite set) decay exponentially. In some regimes, the same is true in higher dimensions. It is however not clear whether the eigenfunctions have an exact rate of exponential decay. In the strip, it is natural to expect that the rate should be given by the slowest Lyapunov exponent, however, only the upper bound has been previously established. We shall discuss some recent progress on this problem, and its connection to a question, perhaps interesting in its own right, from the theory of random matrix products. Based on joint work with Ilya Goldsheid.
ThursdayMar 05, 202013:30
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:Emanuel Milman (Technion) & Tal Orenshtein (TU Berlin)Title:DOUBLE SEMINARAbstract:opens in new windowin html    pdfopens in new window

Speaker #1: Emanuel Milman (Technion)

Title: Functional Inequalities on sub-Riemannian manifolds via QCD

Abstract:We are interested in obtaining Poincar\'e and log-Sobolev inequalities on domains in sub-Riemannian manifolds (equipped with their natural sub-Riemannian metric and volume measure).
It is well-known that strictly sub-Riemannian manifolds do not satisfy any type of Curvature-Dimension condition CD(K,N), introduced by Lott-Sturm-Villani some 15 years ago, so we must follow a different path. We show that while ideal (strictly) sub-Riemannian manifolds do not satisfy any type of CD condition, they do satisfy a quasi-convex relaxation thereof, which we name QCD(Q,K,N). As a consequence, these spaces satisfy numerous functional inequalities with exactly the same quantitative dependence (up to a factor of Q) as their CD counterparts. We achieve this by extending the localization paradigm to completely general interpolation inequalities, and a one-dimensional comparison of QCD densities with their "CD upper envelope". We thus obtain the best known quantitative estimates for (say) the L^p-Poincar\'e and log-Sobolev inequalities on domains in the ideal sub-Riemannian setting, which in particular are independent of the topological dimension. For instance, the classical Li-Yau / Zhong-Yang spectral-gap estimate holds on all Heisenberg groups of arbitrary dimension up to a factor of 4.
No prior knowledge will be assumed, and we will (hopefully) explain all of the above notions during the talk.

Speaker #2: Tal Orenshtein (TU Berlin)

Title: Rough walks in random environment
Abstract. In this talk we shall review scaling limits for random walks in random environment lifted to the rough path space to the enhanced Brownian motion. Except for the immediate application to SDEs, this adds some new information on the structure of the limiting path. Time permitted, we shall elaborate on the tools to tackle these problems. Based on joint works with Olga Lopusanschi, with Jean-Dominique Deuschel and Nicolas Perkowski and with Johaness Bäumler, Noam Berger and Martin Slowik.

ThursdayFeb 13, 202013:30
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:Anirban Basak Title:Upper tail large deviations of subgraph counts in sparse random graphsAbstract:opens in new windowin html    pdfopens in new window

For a $\Delta$-regular connected graph ${\sf H}$ the problem of determining the upper tail large deviation for the number of copies of ${\sf H}$ in an Erd\H{o}s-R\'{e}nyi graph on $n$ vertices with edge probability $p$ has generated significant interests. In the sparse regime, i.e. for $p \ll 1$, when $np^{\Delta/2} \gg (\log n)^{1/(v_{\sf H}-2)}$, where $v_{\sf H}$ is the number of vertices in ${\sf H}$, the upper tail large deviation event is believed to occur due to the presence of localized structures. Whereas, for $p$ below the above threshold the large deviation is expected to be given by that of a Poisson random variable. In this talk, we will discuss some progress in resolving this conjecture.

This is based on joint work with Riddhipratim Basu.

ThursdayJan 30, 202013:30
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:Amir Yehudayoff Title:Anti-concentration of inner productsAbstract:opens in new windowin html    pdfopens in new window

The plan is to discuss anti-concentration of the inner product between two independent random vectors in Euclidean space. We shall go over some background and applications, and see some proofs.

ThursdayJan 23, 202013:30
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:James Lee Title:Metric entropies, L_1 transportation, and competitive analysisAbstract:opens in new windowin html    pdfopens in new window

The MTS problem (Borodon, Linial, and Saks 1992) is a general model for the analysis of algorithms that optimize in the presence of information arriving over time, where the state space is equipped with a metric. I will discuss a relatively new approach to this area, where both the algorithm and method of analysis are derived canonically from a choice of a "regularizer" on the probability simplex. The regularizer can be interpreted as a Riemannian structure on the simplex, and then the algorithm is simply a gradient flow.

Defining the regularizer as an appropriate "noisy" multiscale metric entropy (similar to, e.g., the Talagrand \gamma_1 functional) yields the best-known competitive ratio for every metric space. For ultrametrics (aka, "HST metrics"), this achieves the conjectured bound, but the algorithm falls short of resolving the MTS conjecture for many other spaces, including subsets of the real line.

ThursdayJan 09, 202013:30
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:Abel FarkasTitle:The Geometric measure theory of the Brownian pathAbstract:opens in new windowin html    pdfopens in new window

Let B denote the range of the Brownian motion in R^d. For a deterministic Borel a measure nu we wish to find a random measure mu such that the support of mu is contained in B and the expectation of mu is nu. We discuss when exactly can there be such a random measure and construct in those cases. We establish a formula for the expectation of the double integral with respect to mu, which is a strong tool for the geometric measure theory of the Brownian path.

ThursdayJan 02, 202013:30
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:Dima Dolgopyat Title:Edgeworth expansion in Central Limit TheoremAbstract:opens in new windowin html    pdfopens in new window

It is well known that Central Limit Theorem is very effective giving a reasonable approximation for sums of a quite small number of terms. Edgeworth expansions provide a convenient way to control the error in the Central Limit Theorem. In this talk I will review some recent results on this subject related to the interface between probability and dynamics.  

ThursdayJan 02, 202013:30
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:Dima Dolgopyat Title:Edgeworth expansion in Central Limit TheoremAbstract:opens in new windowin html    pdfopens in new window

It is well known that Central Limit Theorem is very effective giving a reasonable approximation for sums of a quite small number of terms. Edgeworth expansions provide a convenient way to control the error in the Central Limit Theorem. In this talk I will review some recent results on this subject related to the interface between probability and dynamics.  

ThursdayDec 26, 201913:30
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:Max Fathi Title:A new proof of the Caffarelli contraction theoremAbstract:opens in new windowin html    pdfopens in new window

The Caffarelli contraction theorem states that the Brenier optimal transport map sending the Gaussian measure onto a uniformly log-concave probability measure is lipschitz. In this talk, I will present a new proof, using entropic regularization and a variational characterization of lipschitz transport maps due to Gozlan and Juillet. Based on joint work with Nathael Gozlan and Maxime Prod'homme.

ThursdayDec 19, 201913:30
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:Esty Kelman & Asaf KatzTitle:DOUBLE SEMINARAbstract:opens in new windowin html    pdfopens in new windowDOUBLE SEMINAR

Esty Kelman [TAU]   Title: Boolean Functions and Their Effective Degree

Abstract: The KKL Theorem (there is always a significantly influential variable) is sometimes tight as in the Tribes functions, but many times is not tight as in the Majority function. We propose a generalized version of KKL, with a new parameter - the effective degree, which replaces the role of the average degree in the KKL statement. This allows us to prove a tight result for many cases shows how large the influence of the most influential variable is. We generalize KKL in another manner finding a significantly influential variable within a subset of variables. This generalization, in turn, implies a generalized version of Friedgut Junta Theorem. If time allows, we will see how easily our Theorem implies the Friedgut Junta Theorem and stronger Junta results.

 

Asaf Katz [U Chicago]  Title: Measure rigidity for Anosov flows via the factorization method

Abstract: We show how the factorization method, pioneered by Eskin and Mirzakhani in their groundbreaking work about measure classification for the moduli space of translation surfaces, can be adapted to smooth ergodic theory.
Using this adaption, we show that for a quantitatively non-integrable Anosov flow, every generalized u-Gibbs measure is absolutely continuous with respect to the whole unstable manifold.

 

ThursdayDec 12, 201913:30
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:Naomi Feldheim Title:Existence of persistence exponent for Gaussian stationary functionsAbstract:opens in new windowin html    pdfopens in new window

Let Z(t) be a Gaussian stationary function on the real line, and fix a level L>0.

We are interested in the asymptotic behavior of the persistence probability: P(T) = P( Z(t) > L, for all t in [0,T] ).
One would guess that for "nice processes", the behavior of P(T) should be exponential. For non-negative correlations this may be established via sub-additivity arguments. However, so far, not a single example with sign-changing correlations was known to exhibit existence of the limit of {Log P(T)}/T, as T approaches infinity (that is, to have a true "persistence exponent").
In the talk I will present a proof of existence of the persistence exponent, for processes whose spectral measure is monotone on [0,∞) and is continuous and non-vanishing at 0. This includes, for example, the sinc-kernel process (whose covariance function is sin(t)/t ).
Joint work with Ohad Feldheim and Sumit Mukherjee.

ThursdayNov 28, 201913:30
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:Amir Dembo & Alexander Fish Title:DOUBLE SEMINARAbstract:opens in new windowin html    pdfopens in new window

Amir Dembo (Stanford) "Dynamics for spherical spin glasses: Disorder dependent initial conditions"

Abstract: In this talk, based on a joint work with Eliran Subag, I will explain how to rigorously derive the integro-differential equations that arise in the thermodynamic limit of the empirical correlation and response functions for Langevin dynamics in mixed spherical p-spin disordered mean-field models.
I will then compare the large time asymptotic of these equations in case of a uniform (infinite-temperature) starting point, to what one obtains when starting within one of the spherical bands on which the Gibbs measure  concentrates at low temperature, commenting on the existence of an aging phenomenon, and on the relations with the recently discovered geometric structure of the Gibbs measures at low temperature.

Alexander Fish (Sydney)  "Finite configurations in trees of positive growth rate"

Abstract: We will talk on the relation between the abundance of finite configurations that we observe in trees and their growth rate.
We will survey the Furstenberg-Weiss correspondence principle which relates a tree of positive growth rate with Markov process, and subsequently provides a quantitative relationship between the density of a finite subtree that appears in the tree with a quantity defined on the Markov space and which can be estimated by use of ergodic methods. We will sketch a proof of one direct and one inverse theorem relating the abundance of certain finite configurations and the growth rate of a tree. Some open problems related to this work will be discussed. Based on a joint work with Leo Jiang (Toronto).

ThursdayNov 14, 201913:30
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:Jonathan Hermon Title:Random Cayley graphsAbstract:opens in new windowin html    pdfopens in new window

We consider the random Cayley graph of a finite group $G$ formed by picking $k$ random generators uniformly at random:

  1. We prove universality of cutoff (for the random walk) and a concentration of measure phenomenon in the Abelian setup (namely, that all but $o(|G|)$ elements lie at distance $[R-o(R),R-o(R)]$ from the origin, where $R$ is the minimal ball in $Z^k$ of size at least $|G|$), provided $k \gg 1$ is large in terms of the size of the smallest generating set of $G$. As conjectured by Aldous and Diaconis, the cutoff time is independent of the algebraic structure (it is given by the time at which the entropy of a random walk on $Z^k$ is $\log|G|$).
  2. We prove analogous results for the Heisenberg $H_{p,d}$  groups  of  $d \times d$ uni-upper triangular matrices with entries defined mod $p$, for $p$ prime and $d$ fixed or diverging slowly.
  3.  Lastly, we resolve a conjecture of D. Wilson that if $G$ is a group of size at most $2^d$ then for all $k$ its mixing time in this model is as rapid as that of $Z_2^d$ and likewise, that the slowest mixing $p$-group of a given size is $Z_p^d$.

(Joint work with Sam Thomas.)

ThursdayOct 31, 201913:30
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:Mikhail Ostrovskii Title:Geometry of transportation cost (a.k.a. Earth Mover or Wasserstein distance)Abstract:opens in new windowin html    pdfopens in new window

We consider (finitely supported) transportation problems on a metric space M. They form a vector space TP(M). The optimal transportation cost for such transportation problems is a norm on this space. This normed space is of interest for the theory of metric embeddings because the space M embeds into it isometrically. I am going to talk about geometry of such normed spaces. The most important questions for this talk are relations of these spaces with $L_1$ and $L_\infty$ spaces.

ThursdayOct 24, 201913:30
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:Sergey G. BobkovTitle:Transport bounds for empirical measuresAbstract:opens in new windowin html    pdfopens in new window

We will be discussing a Fourier-analytic approach to optimal matching between independent samples, with an elementary proof of the Ajtai-Komlos-Tusnady theorem.
The talk is based on a joint work with Michel Ledoux.  

ThursdayJul 25, 201913:30
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:Jonathan Hermon Title:Anchored expansion in supercritical percolation on nonamenable graphs.Abstract:opens in new windowin html    pdfopens in new window

Let G be a transitive nonamenable graph, and consider supercritical Bernoulli bond percolation on G. We prove that the probability that the origin lies in a finite cluster of size n decays exponentially in n. We deduce that:

  1. Every infinite cluster has anchored expansion almost surely. This answers positively a question of Benjamini, Lyons, and Schramm (1997).
  2. Various observables, including the percolation probability and the truncated susceptibility are analytic functions of p throughout the entire supercritical phase.

Joint work with Tom Hutchcroft.

ThursdayJul 11, 201913:30
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:Shamgar GurevitchTitle:Harmonic Analysis on GL_n over finite fieldsAbstract:opens in new windowin html    pdfopens in new window

There are many formulas that express interesting properties of a finite group G in terms of sums over its characters. For evaluating or estimating these sums, one of the most salient quantities to understand is the {\it character ratio}:

$$

trace(\rho(g))/dim(\rho),

$$

for an irreducible representation $\rho$ of G and an element g of G. For example, Diaconis and Shahshahani stated a formula of the mentioned type for analyzing certain random walks on G.

Recently, we discovered that for classical groups G over finite fields there is a natural invariant of representations that provides strong information on the character ratio. We call this invariant rank. This talk will discuss the notion of rank for GLn over finite fields, and explain how one can apply the results to verify mixing time and rate for certain random walks.

The talk will assume basic notions of linear algebra in Hilbert spaces, and the definition of a group.

This is joint work with Roger Howe (Yale and Texas AM).

ThursdayJul 04, 201913:30
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:Eviatar Procaccia Title:Stationary Hastings-Levitov modelAbstract:opens in new windowin html    pdfopens in new window

We construct and study a stationary version of the Hastings-Levitov(0) model. We prove that unlike the classical model, in the stationary case particle sizes are constant in expectation, yielding that this model can be seen as a tractable off-lattice Diffusion Limited Aggregation (DLA). The stationary setting together with a geometric interpretations of the harmonic measure yields new geometric results such as stabilization, finiteness of arms and unbounded width in mean of arms. Moreover we can present an exact conjecture for the fractal dimension.

ThursdayJun 20, 201913:30
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:Mark Rudelson (UMich) & Serguei Popov (IMECC)Title:Double seminar Abstract:opens in new windowin html    pdfopens in new window

Speaker 1: Mark Rudelson (UMich)

Title: Circular law for sparse random matrices.

Abstract: Consider a sequence of $n$ by $n$ random matrices $A_n$ whose entries are independent identically distributed random variables. The circular law asserts that the distribution of the eigenvalues of properly normalized matrices $A_n$ converges to the uniform measure on the unit disc as $n$ tends to infinity. We prove this law for sparse random matrices under the optimal sparsity assumption. Joint work with Konstantin Tikhomirov.

Speaker 2: Serguei Popov (IMECC)

Title: On the range of a two-dimensional conditioned random walk

Abstract: We consider the two-dimensional simple random walk conditioned on never hitting the origin. This process is a Markov chain, namely, it is the Doob $h$-transform of the simple random walk with respect to the potential kernel. It is known to be transient and we show that it is "almost recurrent'' in the sense that each infinite set is visited infinitely often, almost surely. We prove that, for a "typical large set", the proportion of its sites visited by the conditioned walk is approximately a Uniform$[0,1]$ random variable. Also, given a set $G\subset\R^2$ that does not "surround" the origin, we prove that a.s.\ there is an infinite number of $k$'s such that $kG\cap \Z^2$ is unvisited. These results suggest that the range of the conditioned walk has "fractal" behavior. This is a joint work with Nina Gantert and Marina Vachkovskaia, see arxiv.org/abs/1804.00291 Also, there is much more about conditioned walks in my new book (www.ime.unicamp.br/~popov/2srw.pdf, work in progress). Comments and suggestions on the latter are very welcome!

ThursdayJun 06, 201913:30
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:Yotam Smilansky Title:Statistics of colored Kakutani sequences of partitionsAbstract:opens in new windowin html    pdfopens in new window

We consider statistical questions concerning colored sequences of partitions, produced by applying a partition process which was first introduced by Kakutani for the 1-dimensional case. This process can be generalized as the application of a fixed multiscale substitution rule, defined on a finite set of colored sets in R^d, on elements of maximal measure in each partition. Colored sets appearing in the sequence are modeled by certain flows on an associated directed weighted graph, and natural statistical questions can be reformulated as questions on the distribution of paths on graphs. Under some incommensurability assumptions, we show that special properties of Laplace transforms of the relevant counting functions imply explicit statistical results

ThursdayMay 23, 201913:30
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:Gregory Berkolaiko Title:Nodal statistics of graph eigenfunctionsAbstract:opens in new windowin html    pdfopens in new window

Understanding statistical properties of zeros of Laplacian eigenfunctions is a program which is attracting much attention from mathematicians and physicists. We will discuss this program in the setting of "quantum graphs", self-adjoint differential operators acting on functions living on a metric graph. Numerical studies of quantum graphs motivated a conjecture that the distribution of nodal surplus (a suitably rescaled number of zeros of the n-th eigenfunction) has a universal form: it approaches Gaussian as the number of cycles grows. The first step towards proving this conjecture is a result established for graphs which are composed of cycles separated by bridges. For such graphs we use the nodal-magnetic theorem of the speaker, Colin de Verdiere and Weyand to prove that the distribution of the nodal surplus is binomial with parameters p=1/2 and n equal to the number of cycles. Based on joint work with Lior Alon and Ram Band.

ThursdayApr 18, 201913:30
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:Michał Strzelecki Title:On modified log-Sobolev inequalitiesAbstract:opens in new windowin html    pdfopens in new window

In order to prove concentration estimates for (products of) measures with heavier tails than the standard Gaussian measure one can use several variants of the classical log-Sobolev inequality, e.g., Beckner-type inequalities of Latala and Oleszkiewicz or modified log-Sobolev inequalities of Gentil, Guillin, and Miclo. The main result I plan to present asserts that a probability measure on R^d which satisfies the former inequality satisfies also the latter. Based on joint work with Franck Barthe.

ThursdayApr 11, 201913:30
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:Lisa Hartung Title:From 1 to 6 in branching Brownian motionAbstract:opens in new windowin html    pdfopens in new window

 Brownian motion is a classical process in probability theory belonging to the class of  "Log-correlated random fields". It is well known do to Bramson that the order of the maximum has a different logarithmic correction as the corresponding independent setting.
In this talk we look at a version of branching Brownian motion where we slightly vary the diffusion parameter in a way that, when looking at the order of the maximum, we can smoothly interpolate between the logarithmic correction for independent random variables ($\frac{1}{2\sqrt 2}\ln(t)$) and the logarithmic correction of BBM ($\frac{3}{2\sqrt 2}\ln(t)$) and the logarithmic correction of 2-speed BBM with increasing variances ($\frac{6}{2\sqrt 2}\ln(t)$). We also establish in all cases the asymptotic law of the maximum and characterise the extremal process, which turns out to coincide essentially with that of standard BBM. We will see that the key to the above results is a precise understanding of the entropic repulsion experienced by an extremal particle. (joint work with A. Bovier)

ThursdayMar 28, 201913:30
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:Ami Viselter Title:Convolution semigroups and generating functionals on quantum groupsAbstract:opens in new windowin html    pdfopens in new window

The theory of locally compact quantum groups grew out of the need to extend Pontryagin's duality for locally compact abelian groups to a wider class of objects, as well as from a modern "quantum" point of view suggesting the replacement of some algebras of functions on a group by non-commutative objects, namely operator algebras. In this talk, which will be split into two parts, we will show how several fundamental notions from probability and geometric group theory fit in this framework.

The first part will be an introduction to locally compact quantum groups. We will present the rationale and the definitions, give examples, and explain how the theory is related to other branches of math. If time permits, we will also touch upon more specific notions related to the second part.

In the second part we will discuss convolution semigroups of states, as well as generating functionals, on locally compact quantum groups. One type of examples comes from probability: the family of distributions of a L\'evy process form a convolution semigroup, which in turn admits a natural generating functional. Another type of examples comes from (locally compact) group theory, involving semigroups of positive-definite functions and conditionally negative-definite functions, which provide
important information about the group's geometry. We will explain how these notions are related and how all this extends to the quantum world; derive geometric characterizations of two approximation properties of locally compact quantum groups; see how generating functionals may be (re)constructed and study their domains; and indicate how our results can be used to study cocycles.

Based on joint work with Adam Skalski.

No background in operator algebras will be assumed.

ThursdayMar 21, 201913:30
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:Zemer Kosloff Title:On the local limit theorem in dynamical systemsAbstract:opens in new windowin html    pdfopens in new window
In 1987, Burton and Denker proved the remarkable result that in every aperiodic dynamical systems (including irrational rotations for example) there is a square integrable, zero mean function such that its corresponding time series satisfies a CLT. Subsequently, Volny showed that one can find a function which satisfies the strong (almost sure) invariance principle. All these constructions resulted in a non-lattice distribution. In a joint work with Dalibor Volny we show that there exists an integer valued cocycle which satisfies the local limit theorem. The first hour will involve painting (Rokhlin towers) while the second one will be mainly concerned with the proof of the local CLT.
ThursdayMar 14, 201913:30
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:Marta Strzelecka Title:On k-maxima of log-concave vectorsAbstract:opens in new windowin html    pdfopens in new window

We establish two-sided bounds for expectations of order statistics (k-th maxima) of moduli of coordinates of centred log-concave random vectors with uncorrelated coordinates. Our bounds are exact up to multiplicative universal constants in the unconditional case for all k and in the isotropic case for k < c n^{5/6}. We also derive two-sided estimates for expectations of sums of k largest moduli of coordinates for some classes of random vectors. The talk will be based on the joint work with Rafal Latala.  

ThursdayMar 07, 201913:30
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:Ori Gurel-Gurevich Title:Random walks on planar graphs Abstract:opens in new windowin html    pdfopens in new window

We will discuss several results relating the behavior of a random walk on a planar graph and the geometric properties of a nice embedding of the graph in the plane (specifically, circle packing of the graph). One such a result is that for a bounded degree graph, the simple random walk is recurrent if and only if the boundary of the nice embedding is a polar set (that is, Brownian motion misses it almost surely). If the degrees are unbounded, this is no longer true, but for the case of circle packing of a triangulation, there are weights which are obtained naturally from the circle packing, such that when the boundary is polar, the weighted random walk is recurrent (we believe the converse also hold). These weights arise also in the context of discrete holomorphic and harmonic functions, a discrete analog of complex holomorphic functions. We show that as the sizes of circles, or more generally, the lengths of edges in the nice embedding of the graph tend to zero, the discrete harmonic functions converge to their continuous counterpart with the same boundary conditions. Equivalently, that the exit measure of the weighted random walk converges to the exit measure of standard Brownian motion. This improves previous results of Skopenkov 2013 and Werness 2015, who proves similar results under additional local and global assumptions on the embedding. In particular, we make no assumptions on the degrees of the graph, making the result applicable to models of random planar maps.

Based of joint works with Daniel Jerison, Asaf Nachmias, Matan Seidel and Juan Souto.

ThursdayFeb 28, 201913:30
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:Fanny Augeri (WIS) + Elliot Paquette (Ohio)Title:DOUBLE SEMINARAbstract:opens in new windowin html    pdfopens in new windowDOUBLE SEMINAR

Speaker 1: Fanny Augeri (WIS)
Title : Nonlinear large deviations bounds with applications to sparse Erdos-Renyi graphs.
Abstract: In this talk, I will present the framework of the so-called nonlinear large deviations introduced by Chatterjee and Dembo. In a seminal paper, they provided a sufficient criterion in order that the large deviations of a function on the discrete hypercube to be due by only changing the mean of the background measure. This sufficient condition was formulated in terms of the complexity of the gradient of the function of interest. I will present general nonlinear large deviation estimates similar to Chatterjee-Dembo's original bounds except that we do not require any second order smoothness. The approach relies on convex analysis arguments and is valid for a broad class of distributions. Then, I will detail an application of this nonlinear large deviations bounds to the problem of estimating the upper tail of cycles counts in sparse Erdos-Renyi graphs down to the connectivity parameter $n^{-1/2}$.

Speaker 2: Elliot Paquette (Ohio)
Title: The Gaussian analytic function is either bounded or covers the plane
Abstract: The Gaussian analytic function (GAF) is a power series with independent Gaussian coefficients. In the case that this power series has radius of convergence 1, under mild regularity assumptions on the coefficients, it is a classical theorem that the power series is a.s. bounded on open disk if and only if it extends continuously to a function on the closed unit disk a.s. Nonetheless, there exists a natural range of coefficients in which the GAF has boundary values in L-p, but is a.s. unbounded. How wild are these boundary values? Well, Kahane asked if a GAF either a.s. extends continuously to the closed disk or a.s. has range covering the whole plane. We show partial progress in establishing this in the affirmative.
Joint with Alon Nishry.

ThursdayFeb 21, 201913:30
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:Houcein Abdalaoui Title:On Banach problem and simple Lebesgue spectrumAbstract:opens in new windowin html    pdfopens in new window

Following Ulam, Banach asked on the existence of dynamical system on real line with simple Lebesgue spectrum. I discuss the connection of this problem to the famous Banach problem in ergodic theory due essentially to Rohklin. I will further present my recent contribution to this problem and the connection to so called Erdos flat polynomials problem in Harmonic analysis due to J. Bourgain . My talk is based on my recent work and joint work with Mahendra Nadkarni (CBS Mumbai, India).

ThursdayFeb 14, 201913:30
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:Rafael Butez Title:On the extremal particles of a Coulomb gas and random polynomialsAbstract:opens in new windowin html    pdfopens in new window

The purpose of this talk is to understand the behavior of the extremal zeros of random polynomials of the form $ P_N(z) = \sum_{k=0}^{N} a_k R_k(z)$ where the family $(R_k)_{k \leq N}$ is an orthonormal basis for the scalar product $\langle P,Q \rangle = \int P(z) \overline{Q(z)} e^{-2N V^{\nu(z)}} d\nu(z)$ with $\nu$ a radial probability measure on $\CC$ and $V^{\nu}(z)= \int \log |z-w|d\nu(w)$.

Although the zeros of these polynomials spread like the measure $\nu$, the zeros of maximum modulus lie outsite of the support. More precisely, the point process of the roots outside of the support of the equilibrium measure converges towards the Bergman point process of the complement of the support.

We also study similar results on a model of Coulomb gases in dimension $2$ where the confining potential is generated by the presence of a fixed background of positive charges. If $\nu$ is a probability measure, we study the system of particles with joint distribution on $\CC^N$, $\frac{1}{Z_N} \prod_{i \leq j} |x_i-x_j|^2 e^{-2(N+1)\sum_{k=1}^{N}V^{\nu}(x_k)} d\ell_{\CC^N}(x_1,\dots,x_N).$ This model in closely related to the study of the zeros of random polynomials. We show that the extremal particles of this Coulomb gas present a similar behavior to the random polynomial setting.

All the results mentioned above are done in collaboration with David Garcia-Zelada.

ThursdayFeb 07, 201913:30
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:Tsviqa Lakrec Title:Scenery Reconstruction for a Random Walk on Random Scenery with Adversarial Error InsertionAbstract:opens in new windowin html    pdfopens in new window

Consider a simple random walk on $\mathbb{Z}$ with a random coloring of $\mathbb{Z}$. Look at the sequence of the first $N$ steps taken and colors of the visited locations. From it, you can deduce the coloring of approximately $\sqrt{N}$ integers. Suppose an adversary may change $\delta N$ entries in that sequence. What can be deduced now? We show that for any $\theta<0.5,p>0$, there are $N_{0},\delta_{0}$ such that if $N>N_{0}$ and $\delta<\delta_{0}$ then with probability $>1-p$ we can reconstruct the coloring of $>N^{\theta}$ integers.

ThursdayJan 31, 201913:30
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:Dan Mikulincer Title:Quantitative high-dimensional CLTs via martingale embeddingsAbstract:opens in new windowin html    pdfopens in new window

We introduce a new method for obtaining quantitative convergence rates for the central limit theorem (CLT) in the high dimensional regime. The method is based on the notion of a martingale embedding, a multivariate analogue of Skorokhod's embedding. Using the method we are able to obtain several new bounds for convergence in transportation distance and in entropy, and in particular: (a) We improve the best known bound, for convergence in quadratic Wasserstein transportation distance for bounded random vectors; (b) We derive a non-asymptotic convergence rate for the entropic CLT in arbitrary dimension, for log-concave random vectors; (c) We give an improved bound for convergence in transportation distance under a log-concavity assumption and improvements for both metrics under the assumption of strong log-concavity.

In this talk, we will review the method, and explain how one might use it in order to prove quantitative statements about rates of convergence.

ThursdayJan 24, 201913:30
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:Noam Lifshhitz Title:Sharp thresholds for sparse functions with applications to extremal combinatorics. Abstract:opens in new windowin html    pdfopens in new window

The sharp threshold phenomenon is a central topic of research in the analysis of Boolean functions. Here, one aims to give sufficient conditions for a monotone Boolean function f to satisfy $\mu p(f)=o(\mu q(f))$, where $q = p + o(p)$, and $\mu p(f)$ is the probability that $f=1$ on an input with independent coordinates, each taking the value $1$ with probability $p$. The dense regime, where $\mu p(f)=\Theta(1)$, is somewhat understood due to seminal works by Bourgain, Friedgut, Hatami, and Kalai. On the other hand, the sparse regime where $\mu p(f)=o(1)$ was out of reach of the available methods. However, the potential power of the sparse regime was envisioned by Kahn and Kalai already in 2006. In this talk we show that if a monotone Boolean function $f$ with $\mu p(f)=o(1)$ satisfies some mild pseudo-randomness conditions then it exhibits a sharp threshold in the interval $[p,q]$, with $q = p+o(p)$. More specifically, our mild pseudo-randomness hypothesis is that the $p$-biased measure of $f$ does not bump up to $\Theta(1)$ whenever we restrict $f$ to a sub-cube of constant co-dimension, and our conclusion is that we can find $q=p+o(p)$, such that $\mu p(f)=o(\mu q(f))$ At its core, this theorem stems from a novel hypercontactive theorem for Boolean functions satisfying pseudorandom conditions, which we call `small generalized influences'. This result takes on the role of the usual hypercontractivity theorem, but is significantly more effective in the regime where $p = o(1)$. We demonstrate the power of our sharp threshold result by reproving the recent breakthrough result of Frankl on the celebrated Erdos matching conjecture, and by proving conjectures of Huang--Loh--Sudakov and Furedi--Jiang for a new wide range of the parameters. Based on a joint work with Keevash, Long, and Minzer.

ThursdayJan 17, 201913:30
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:Eliran Subag Title:Optimization of random polynomials on the sphere in the full-RSB regimeAbstract:opens in new windowin html    pdfopens in new window

To compute the spectral norm of a p-tensor one needs to optimize a homogeneous polynomial of degree p over the sphere. When p=2 (the matrix case) it is algorithmically easy, but for p>2 it can be NP-hard. In this talk I will focus on (randomized) optimization in high-dimensions when the objective function is a linear combination of homogeneous polynomials with Gaussian coefficients. Such random functions are called spherical spin glasses in physics and have been extensively studied since the 80s. I will describe certain geometric properties of spherical spin glasses unique to the full-RSB case, and explain how they can be used to design a polynomial time algorithm that finds points within a small multiplicative error from the global maximum.

ThursdayJan 10, 201913:30
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:Godofredo Iommi and Paul DarioTitle:Double SeminarAbstract:opens in new windowin html    pdfopens in new window

Godofredo Iommi (PUC Chile)

Title: Upper semi-continuity of the entropy map for Markov shifts
Abstract: In this talk I will show that for finite entropy countable Markov shifts the entropy map is upper semi-continuous when restricted to the set of ergodic measures. This is joint work with Mike Todd and Anibal Velozo.

Paul Dario (ENS)

Title: Homogenization on supercritical percolation cluster
Abstract: The standard theory of stochastic homogenization requires an assumption of uniform ellipticity on the environment. In this talk, we investigate how one can remove this assumption in a specific case: the infinite cluster of the supercritical Bernouilli percolation of Zd. We will present a renormalization argument for the infinite cluster and show how one can use it to adapt the theory developped in the uniformly elliptic setting. We will then review some results which can be obtained through this technique: homogenization theorem, large scale regularity, Liouville theorem and bounds on the corrector.

ThursdayJan 03, 201913:30
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:Chaim Even-Zohar Title:Patterns in Random PermutationsAbstract:opens in new windowin html    pdfopens in new window

Every $k$ entries in a permutation can have one of k! different relative orders, called patterns. How many times does each pattern occur in a large random permutation of size $n$? The distribution of this k!-dimensional vector of pattern densities was studied by Janson, Nakamura, and Zeilberger (2015). Their analysis showed that some component of this vector is asymptotically multinormal of order $1/\sqrt(n)$, while the orthogonal component is smaller. Using representations of the symmetric group, and the theory of U-statistics, we refine the analysis of this distribution. We show that it decomposes into $k$ asymptotically uncorrelated components of different orders in $n$, that correspond to representations of $S_k$. Some combinations of pattern densities that arise in this decomposition have interpretations as practical nonparametric statistical tests.

ThursdayDec 27, 201813:30
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:Yury Makarychev Title:Performance of Johnson-Lindenstrauss Transform for k-Means and k-Medians ClusteringAbstract:opens in new windowin html    pdfopens in new window

Consider an instance of Euclidean k-means or k-medians clustering. We show that the cost of the optimal solution is preserved up to a factor of $(1+\epsilon)$ under a projection onto a random $O(log(k/\epsilon)/\epsilon^2)$-dimensional subspace whp. Further, the cost of every clustering is preserved within $(1+\epsilon)$. Crucially, the dimension does not depend on the total number of points n in the instance. Additionally, our result applies to Euclidean k-clustering with the distances raised to the p-th power for any constant $p$.

For k-means, our result resolves an open problem posed by Cohen, Elder, Musco, Musco, and Persu (STOC 2015); for k-medians, it answers a question raised by Kannan.

Joint work with Konstantin Makarychev and Ilya Razenshteyn.

ThursdayDec 20, 201813:30
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:Max Fathi Title:Stability in the Bakry-Emery theoremAbstract:opens in new windowin html    pdfopens in new window

The Bakry-Emery theorem asserts that uniformly log-concave probability measures satisfy certain functional inequalities, with constants that are better than those associated with the Gaussian measure. In this talk, I will explain how if the constant is almost that of the Gaussian, then the measure almost splits off a Gaussian factor, with explicit quantitative bounds. The proof is based on a combination of Stein's method and simple arguments from calculus of variations. Joint work with Thomas Courtade.

ThursdayDec 13, 201813:30
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:Manuel Stadlbauer Title:Exponential decay of quotients of Ruelle operatorsAbstract:opens in new windowin html    pdfopens in new window

Ruelle's operator theorem states that the Ruelle operator $L$, which is a positive operator acting on Holder functions, is conjugated to $P+R$ where $R$ is a one-dimensional projection and the norm of $R$ is smaller than 1. This decomposition, also known as spectral gap, is of interest as it allows to characterise the underlying dynamical system through, e.g., central limit theorems or continuous response to perturbations. However, the conjugation depends on the existence of a positive eigenfunction of $L$, which might not exist in more general, fibred situations due to purely functorial reasons. A possibility to circumvent this problem is to consider quotients of operators of the form $f \mapsto \frac{L^m(f L^n (1))}{L^{m+n}(1)}.$ In fact, it is possible to provide reasonable conditions such that their dual operators contract the Wasserstein distance exponentially in $m$. The result gives rise, for example, to a law of the iterated logarithm for continued fractions with sequentially restricted entries or a topology on the set of equilibrium states for semigroups of expanding maps. This is joint work with Paulo Varandas and Xuan Zhang.

ThursdayDec 06, 201813:30
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:Boaz SlomkaTitle:Improved bounds for Hadwiger’s covering problem via thin-shell estimatesAbstract:opens in new windowin html    pdfopens in new window

A long-standing open problem, known as Hadwiger's covering problem, asks what is the smallest natural number $N(n)$ such that every convex body in {\mathbb R}^n can be covered by a union of the interiors of at most $N(n)$ of its translates. Despite continuous efforts, the best general upper bound known for this number remains as it was more than sixty years ago, of the order of ${2n \choose n} \ln n$.

In this talk, I will discuss some history of this problem and present a new result in which we improve this bound by a sub-exponential factor. Our approach combines ideas from previous work, with tools from Asymptotic Geometric Analysis. As a key step, we use thin-shell estimates for isotropic log-concave measures to prove a new lower bound for the maximum volume of the intersection of a convex body $K$ with a translate of $-K$. We further show that the same bound holds for the volume of $K\cap(-K)$ if the center of mass of $K$ is at the origin.

If time permits we shall discuss some other methods and results concerning this problem and its relatives.

Joint work with H. Huang, B. Vritsiou, and T. Tkocz

ThursdayNov 29, 201813:30
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:Amir Dembo Title:Large deviations of subgraph counts for sparse random graphsAbstract:opens in new windowin html    pdfopens in new window

For fixed t>1 and L>3 we establish sharp asymptotic formula for the log-probability that the number of cycles of length L in the Erdos - Renyi random graph G(N,p) exceeds its expectation by a factor t,  assuming only that p >> log N/sqrt(N). We obtain such sharp upper tail  bounds also for the Schatten norms of the corresponding adjacency matrices, and in a narrower range of p=p(N), also for general subgraph counts. In this talk, based on a recent joint work with Nick Cook, I will explain our approach and in particular our quantitative refinement of Szemeredi's regularity lemma for sparse random graphs in the large deviations regime.

ThursdayNov 08, 201813:30
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:Snir Ben-Ovadia Title:What are SRB and GSRB measures, and a new characterisation of their existence.Abstract:opens in new windowin html    pdfopens in new window

SRB measures are an important object in dynamical systems and mathematical physics. Named after Sinai , Ruelle and Bowen, these measures have important properties of being equilibrium states which describe chaotic behaviour, yet may also describe the asymptotic of ``observable” events in the phase space. An open and important question, is in what generality do systems admit SRB measures?

We present the notion of generalised SRB measures (GSRB in short), mention some of their important properties, and present a new condition to characterise their existence on a general setup.

The first part of the talk will describe some of the motivation leading to define and to study SRB measures; and so we will define GSRB measures and compare their properties with the properties sought for SRB measures. We will also describe a case study of examples motivating to study GSRB measures. Our new result is a characterisation of systems admitting GSRB measures.

In the second part of the talk, as much as time permits, we will present some key steps in the construction of GSRB measures.

ThursdayOct 25, 201813:30
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:Daniel Dadush Title:Balancing vectors in any normAbstract:opens in new windowin html    pdfopens in new window

In the vector balancing problem, we are given N vectors v_1,..., v_N in an n-dimensional normed space, and our goal is to assign signs to them, so that the norm of their signed sum is as small as possible. The balancing constant of the vectors is the smallest number beta, such that any subset of the vectors can be balanced so that their signed sum has norm at most beta.
The vector balancing constant generalizes combinatorial discrepancy, and is related to rounding problems in combinatorial optimization, and to the approximate Caratheodory theorem. We study the question of efficiently approximating the vector balancing constant of any set of vectors, with respect to an arbitrary norm. We show that the vector balancing constant can be approximated in polynomial time to within factors logarithmic in the dimension, and is characterized by (an appropriately optimized version of) a known volumetric lower bound. Our techniques draw on results from geometric functional analysis and the theory of Gaussian processes. Our results also imply an improved approximation algorithm for hereditary discrepancy.
Joint work with Aleksandar Nikolov, Nicole Tomczak-Jaegermann and Kunal Talwar.

ThursdayAug 30, 201813:30
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:Assaf Naor Title:Coarse (non)Universality of Alexandrov SpacesAbstract:opens in new windowin html    pdfopens in new window

We will show that there exists a metric space that does not admit a coarse embedding into any Alexandrov space of global nonpositive curvature, thus answering a question of Gromov (1993). In contrast, any metric space embeds coarsely into an Alexandorv space of nonnegative curvature. Based on joint works with Andoni and Neiman, and Eskenazis and Mendel.

ThursdayAug 16, 201813:30
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:David Ellis Title:Random graphs with constant r-ballsAbstract:opens in new windowin html    pdfopens in new window

Let F be a fixed infinite, vertex-transitive graph. We say a graph G is `r-locally F' if for every vertex v of G, the ball of radius r and centre v in G is isometric to the ball of radius r in F. For each positive integer n, let G_n be a graph chosen uniformly at random from the set of all unlabelled, n-vertex graphs that are r-locally F. We investigate the properties that the random graph G_n has with high probability --- i.e., how these properties depend on the fixed graph F.

We show that if F is a Cayley graph of a torsion-free group of polynomial growth, then there exists a positive integer r_0 such that for every integer r at least r_0, with high probability the random graph G_n = G_n(F,r) defined above has largest component of size between n^{c_1} and n^{c_2}, where 0 < c_1 < c_2 < 1 are constants depending upon F alone, and moreover that G_n has a rather large automorphism group. This contrasts sharply with the random d-regular graph G_n(d) (which corresponds to the case where F is replaced by the infinite d-regular tree).

Our proofs use a mixture of results and techniques from group theory, geometry and combinatorics.

We obtain somewhat more precise results in the case where F is L^d (the standard Cayley graph of Z^d): for example, we obtain quite precise estimates on the number of n-vertex graphs that are r-locally L^d, for r at least linear in d.

Many intriguing open problems remain: concerning groups with torsion, groups with faster than polynomial growth, and what happens for more general structures than graphs.

This is joint work with Itai Benjamini (WIS).

ThursdayJul 26, 201813:30
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:Eliran Subag Title:Free energy landscapes in spherical spin glassesAbstract:opens in new windowin html    pdfopens in new window

I will describe a new approach to the study of spherical spin glass models via free energy landscapes, defined by associating to interior points of the sphere the free energy computed only over the spherical band around them.
They are closely related to several fundamental objects from spin glass theory: the TAP free energy, pure states decomposition, overlap distribution, and temperature chaos. I will explain some of of those connections.

ThursdayJul 05, 201813:30
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:Jonathan Hermon Title:The exclusion process (usually) mixes faster than independent particles.Abstract:opens in new windowin html    pdfopens in new window
The exclusion process is one of the most basic and best studied processes in the literature on interacting particle systems, with connections to card shuffling and statistical mechanics. It has been one of the major examples driving the study of mixing-times. In the exclusion process on an n-vertex graph we have k black particles and n-k white particles, one per site. Each edge rings at rate 1. When an edge rings, the particles occupying its end-points switch positions. Oliveira conjectured that the order of the mixing time of the process is at most that of the mixing-time of k independent particles. Together with Richard Pymar we verify this up to a constant factor for d-regular (or bounded degree) graphs 1 in various cases: (1) the degree d is at least logarithmic in n, or (2) the spectral-gap of a single walk is small (at most log number of vertices to the power 4) or (3) when the number of particles k is roughly n^a for some constant 0 n^c $) is within a $\log \log n$ factor from Oliveira's conjecture. As applications we get new mixing bounds: (a) $O(\log n \log \log n)$ for expanders, (b) order $ \log (dk) $ for the hypercube ${0,1}^d$ and (c) order $(diameter)^2 \log k $ for vertex-transitive graphs of moderate growth and for the giant component of supercritical percolation on a torus
ThursdayJun 28, 201813:30
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:Omer BobrowskiTitle:Homological connectivity and percolation in random geometric complexesAbstract:opens in new windowin html    pdfopens in new window
Connectivity and percolation are two well studied phenomena in random graphs. In this talk we will discuss higher-dimensional analogues of connectivity and percolation that occur in random simplicial complexes. Simplicial complexes are a natural generalization of graphs, consisting of vertices, edges, triangles, tetrahedra, and higher dimensional simplexes. We will mainly focus on random geometric complexes. These complexes are generated by taking the vertices to be a random point process, and adding simplexes according to their geometric configuration. Our generalized notions of connectivity and percolation use the language of homology - an algebraic-topological structure representing cycles of different dimensions. In this talk we will review some recent progress in characterizing and analyzing these phenomena, as well as describing related phase transitions.
ThursdayJun 14, 201813:30
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:Emanuel Milman Title:The Gaussian Double-Bubble and Multi-Bubble ConjecturesAbstract:opens in new windowin html    pdfopens in new window

The classical Gaussian isoperimetric inequality, established in the 70's independently by Sudakov-Tsirelson and Borell, states that the optimal way to decompose $\mathbb{R}^n$ into two sets of prescribed Gaussian measure, so that the (Gaussian) area of their interface is minimal, is by using two complementing half-planes. This is the Gaussian analogue of the classical Euclidean isoperimetric inequality, and is therefore referred to as the "single-bubble" case.

A natural generalization is to decompose $\mathbb{R}^n$ into $q \geq 3$ sets of prescribed Gaussian measure. It is conjectured that when $q \leq n+1$, the configuration whose interface has minimal (Gaussian) area is given by the Voronoi cells of $q$ equidistant points. For example, for $q=3$ (the "double-bubble"conjecture) in the plane ($n=2$), the interface is conjectured to be a "tripod" or "Y" - three rays meeting at a single point in 120 degree angles. For $q=4$ (the "triple-bubble" conjecture) in $\mathbb{R}^3$, the interface is conjectured to be a tetrahedral cone.

We confirm the Gaussian double-bubble and, more generally, multi-bubble conjectures for all $3 \leq q \leq n+1$. The double-bubble case $q=3$ is simpler, and we will explain why.

None of the numerous methods discovered over the years for establishing the classical $q=2$ case seem amenable to the $q \geq 3$ cases, and our method consists of establishing a Partial Differential Inequality satisfied by the isoperimetric profile. To treat $q > 3$, we first prove that locally minimimal configurations must have flat interfaces, and thus convex polyhedral cells. Uniqueness of minimizers up to null-sets is also established.

This is joint work with Joe Neeman (UT Austin).

ThursdayJun 07, 201813:30
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:Ohad Feldheim (HUJI) and Eviatar Procaccia (Texas A&M) Title:Double seminarAbstract:opens in new windowin html    pdfopens in new window

Ohad Feldheim: Convergence of a quantile admission processes
Abstract: Consider the following stochastic model for a growing set. At time 0 the model consists of the singleton S = {-infty}. At every subsequent time, two i.i.d. samples, distributed according to some distribution D on R, are suggested as candidates for S. If the smaller among the two is closer to at least a fraction of r of the current elements of S (in comparison with the larger one), then it is admitted into S.
How will the distribution of the members of S evolve over time as a function of r and D?
This model was suggested by Alon, Feldman, Mansour, Oren and Tennenholtz as a model for the evolution of an exclusive social group. We'll show that the empirical distribution of the elements of S converges to a (not-necessarily deterministic) limit distribution for any r and D.
This we do by relating the process to a random walk in changing environment. The analysis of this random walk involves various classical exponential concentration inequalities as well as a new general inequality concerning mean and minimum of independent random variables.
Joint work with Naomi Feldheim

Eviatar Procaccia:  Stabilization of Diffusion Limited Aggregation in a Wedge
Abstract: We prove a discrete Beurling estimate for the harmonic measure in a wedge in $\mathbf{Z}^2$, and use it to show that Diffusion Limited Aggregation (DLA) in a wedge of angle smaller than $\pi/4$ stabilizes. This allows to consider the infinite DLA and questions about the number of arms, growth and dimension. I will present some conjectures and open problems.

ThursdayMay 31, 201813:30
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:Bhaswar Bhattacharya (UPenn) and Elliot Paquette (Ohio) Title:Double SeminarAbstract:opens in new windowin html    pdfopens in new window

Bhaswar Bhattacharya: Large Deviation Variational Problems in Random Combinatorial Structures

The upper tail problem in the Erdos-Renyi random graph $G\sim\mathcal{G}_{n,p}$, where every edge is included independently with probability $p$, is to estimate the probability that the number of copies of a graph $H$ in $G$ exceeds its expectation by a factor of $1+\delta$. The arithmetic analog of this problem counts the number of $k$-term arithmetic progressions in a random subset of $\{1, 2, \ldots, N\}$, where every element is included independently with probability $p$. The recently developed framework of non-linear large deviations (Chatterjee and Dembo (2016) and Eldan (2017)) shows that the logarithm of these tail probabilities can be reduced to a natural variational problem on the space of weighted graphs/functions. In this talk we will discuss methods for solving these variational problems in the sparse regime ($p \rightarrow 0$), and show how the solutions are often related to extremal problems in combinatorics. (This is based on joint work with Shirshendu Ganguly, Eyal Lubetzky, Xuancheng Shao, and Yufei Zhao.)

Elliot Paquette: Random matrix point processes via stochastic processes

In 2007, Virag and Valko introduced the Brownian carousel, a dynamical system that describes the eigenvalues of a canonical class of random matrices. This dynamical system can be reduced to a diffusion, the stochastic sine equation, a beautiful probabilistic object requiring no random matrix theory to understand. Many features of the limiting eigenvalue point process, the Sine--beta process, can then be studied via this stochastic process. We will sketch how this stochastic process is connected to eigenvalues of a random matrix and sketch an approach to two questions about the stochastic sine equation: deviations for the counting Sine--beta counting function and a functional central limit theorem.

ThursdayMay 17, 201813:30
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:Ron Peled Title:The fluctuations of random surfacesAbstract:opens in new windowin html    pdfopens in new window

Random surfaces in statistical physics are commonly modeled by a real-valued function phi on a lattice, whose probability density penalizes nearest-neighbor fluctuations. Precisely, given an even function V, termed the potential, the energy H(phi) is computed as the sum of V over the nearest-neighbor gradients of phi, and the probability density of phi is set proportional to exp(-H(phi)). The most-studied case is when V is quadratic, resulting in the so-called Gaussian free field. Brascamp, Lieb and Lebowitz initiated in 1975 the study of the global fluctuations of random surfaces for other potential functions and noted that understanding is lacking even in the case of the quartic potential, V(x)=x^4. We will review the state of the art for this problem and present recent work with Alexander Magazinov which finally settles the question of obtaining upper bounds for the fluctuations for the quartic and many other potential functions.

ThursdayMay 10, 201813:30
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:David Jerison (MIT) and Ron Rosenthal (Technion)Title:DOUBLE TALKAbstract:opens in new windowin html    pdfopens in new window

David Jerison: Localization of eigenfunctions via an effective potential
We discuss joint work with Douglas Arnold, Guy David, Marcel Filoche and Svitlana Mayboroda. Consider the Neumann boundary value problem for the operator $L u = - \mbox{div} (A \nabla u) + Vu$ on a Lipschitz domain $\Omega$ and, more generally, on a manifold with or without boundary. The eigenfunctions of $L$ are often localized, as a result of disorder of the potential $V$, the matrix of coefficients $A$, irregularities of the boundary, or all of the above. In earlier work, Filoche and Mayboroda introduced the function $u$ solving $Lu = 1$, and showed numerically that it strongly reflects this localization. Here, we deepen the connection between the eigenfunctions and this {\em landscape} function $u$ by proving that its reciprocal $1/u$ acts as an {\em effective potential}. The effective potential governs the exponential decay of the eigenfunctions of the system and delivers information on the distribution of eigenvalues near the bottom of the spectrum.

Ron Rosenthal: Eigenvector correlation in the complex Ginibre ensemble
The complex Ginibre ensemble is a non-Hermitian random matrix on C^N with i.i.d. complex Gaussian entries normalized to have mean zero and variance 1=N. Unlike the Gaussian unitary ensemble, for which the eigenvectors are orthogonal, the geometry of the eigenbases of the Ginibre ensemble are not particularly well understood. We will discuss a some results regarding the analytic and algebraic structure of eigenvector correlations in this matrix ensemble. In particular, we uncover an extended algebraic structure which describes the asymptotic behavior (as N goes to infinity) of these correlations. Our work extends previous results of Chalker and Mehlig [CM98], in which the correlation for pairs of eigenvectors was computed. Based on a joint work with Nick Crawford.

ThursdayApr 26, 201813:30
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:Anirban Basak Title:Local weak limits of Ising and Potts measures on locally tree-like graphsAbstract:opens in new windowin html    pdfopens in new window

Consider a sequence of growing graphs converging locally weakly to an infinite (possibly random) tree. As there are uncountably many Ising and Potts Gibbs measures on the limiting tree in the low-temperature regime it is not apriori clear whether the local weak limit of such measures exists and if so, identifying the limit remains a challenge. In this talk, I will describe these limits. The talk is based on joint works with Amir Dembo and Allan Sly.

ThursdayApr 12, 201813:30
Geometric Functional Analysis and Probability SeminarRoom 290C
Speaker:David Ellis (Queen Mary U) Benjamin Fehrman (Max Planck Institute)Title:** Double Seminar **Abstract:opens in new windowin html    pdfopens in new window

Talk 1: David Ellis (Queen Mary U)
Title: The edge-isoperimetric problem for antipodally symmetric subsets of the discrete cube.
Abstract: A major open problem in geometry is to solve the isoperimetric problem for n-dimensional real projective space, i.e. to determine, for each real number V, the minimum possible size of the boundary of a (well-behaved) set of volume V, in n-dimensional real projective space. We study a discrete analogue of this question: namely, among all antipodally symmetric subsets of {0,1}^n of fixed size, which sets have minimal edge-boundary? We obtain a complete answer to the second question. This is joint work with Imre Leader (Cambridge)

Talk 2:  Benjamin Fehrman (Max Planck Institute)
Title: Well-posedness of stochastic porous media equations with nonlinear, conservative noise.
Abstract: In this talk, which is based on joint work with Benjamin Gess, I will describe a pathwise well-posedness theory for stochastic porous media equations driven by nonlinear, conservative noise. Such equations arise in the theory of mean field games, as an approximation to the Dean-Kawasaki equation in fluctuating hydrodynamics, to describe the fluctuating hydrodynamics of a zero range process, and as a model for the evolution of a thin film in the regime of negligible surface tension. Our methods are loosely based on the theory of stochastic viscosity solutions, where the noise is removed by considering a class of test functions transported along underlying stochastic characteristics. We apply these ideas after passing to the equation's kinetic formulation, for which the noise enters linearly and can be inverted using the theory of rough paths.

ThursdayMar 29, 201813:30
Geometric Functional Analysis and Probability SeminarRoom 290C
Speaker:Mark Rudelson (UMich), Yinon Spinka (TAU)Title:*** Double Seminar ***Abstract:opens in new windowin html    pdfopens in new window

Mark Rudelson (UMich)
Title: Invertibility of the adjacency matrices of random graphs.
Abstract: Consider an adjacency matrix of a bipartite, directed, or undirected Erdos-Renyi random graph. If the average degree of a vertex is large enough, then such matrix is invertible with high probability. As the average degree decreases, the probability of the matrix being singular increases, and for a sufficiently small average degree, it becomes singular with probability close to 1. We will discuss when this transition occurs, and what the main reason for the singularity of the adjacency matrix is.
This is a joint work with Anirban Basak.

Yinon Spinka (TAU)
Title: Finitary codings of Markov random fields
Abstract: Let X be a stationary Z^d-process. We say that X can be coded by an i.i.d. process if there is a(deterministic and translation-invariant) way to construct a realization of X from i.i.d. variables associated to the sites of Z^d. That is, if there is an i.i.d. process Y and a measurable map F from the underlying space of Y to that of X, which commutes with translations of Z^d and satisfies that F(Y)=X in distribution. Such a coding is called finitary if, in order to determine the value of X at a given site, one only needs to look at a finite (but random) region of Y.
It is known that a phase transition (existence of multiple Gibbs states) is an obstruction for the existence of such a finitary coding. On the other hand, we show that when X is a Markov random field satisfying certain spatial mixing conditions, then X can be coded by an i.i.d. process in a finitary manner. Moreover, the coding radius has exponential tails, so that typically the value of X at a given site is determined by a small region of Y.
We give applications to models such as the Potts model, proper colorings and the hard-core model.

ThursdayMar 15, 201813:30
Geometric Functional Analysis and Probability SeminarRoom 290C
Speaker:Elie Aidekon Title:Points of infinite multiplicity of a planar Brownian motionAbstract:opens in new windowin html    pdfopens in new window

Points of infinite multiplicity are particular points which the Brownian motion visits infinitely often. Following a work of Bass, Burdzy and Khoshnevisan, we construct and study a measure carried by these points. Joint work with Yueyun Hu and Zhan Shi.

ThursdayMar 08, 201813:30
Geometric Functional Analysis and Probability SeminarRoom 290C
Speaker:Erwin Bolthausen Title:On the high temperature phase in mean-field spin glassesAbstract:opens in new windowin html    pdfopens in new window

We present a new way to derive the replica symmetric solution for the free energy in mean-field spin glasses. Only the Sherrington-Kirpatrick case has been worked out in details, but the method also works in other cases, for instance for the perceptron (work in progress), and probably also for the Hopfield net. The method is closely related to the TAP equations (for Thouless-Anderson-Palmer). It does not give any new results, presently, but it gives a new viewpoint, and it looks to be quite promising. As the TAP equations are widely discussed in the physics literature, also at low temperature, it is hoped that the method could be extended to this case, too. But this is open, and probably very difficult.

ThursdayJan 18, 201813:30
Geometric Functional Analysis and Probability SeminarRoom 290C
Speaker:Oren Louidor (Technion) and Alexander Glazman (Tel Aviv).Title:Double SeminarAbstract:opens in new windowin html    pdfopens in new window

Oren Louidor (Technion)
Title: Dynamical freezing in a spin-glass with logarithmic correlations.
Abstract: We consider a continuous time random walk on the 2D torus, governed by the exponential of the discrete Gaussian free field acting as potential. This process can be viewed as Glauber dynamics for a spin-glass system with logarithmic correlations. Taking temperature to be below the freezing point, we then study this process both at pre-equilibrium and in-equilibrium time scales. In the former case, we show that the system exhibits aging and recover the arcsine law as asymptotics for a natural two point temporal correlation function. In the latter case, we show that the dynamics admits a functional scaling limit, with the limit given by a variant of Kolmogorov's K-process, driven by the limiting extremal process of the field, or alternatively, by a super-critical Liouville Brownian motion. Joint work with A. Cortines, J. Gold and A. Svejda.

Alexander Glazman (Tel Aviv)
Title: Level lines of a random Lipschitz function
Abstract: We consider the uniform distribution on Lipschitz functions on the triangular lattice, i.e. all integer-valued functions which differ by 0 or 1 on any two adjacent vertices. We show that with a positive probability such a function exhibits macroscopic level lines. Instead of working directly with Lipschitz functions we map this model to the loop $O(2)$ model with parameter $x=1$. The loop $O(n)$ model is a model for a random collection of non-intersecting loops on the hexagonal lattice, which is believed to be in the same universality class as the spin $O(n)$ model.
A main tool in the proof is a positive association (FKG) property that was recently shown to hold when $n \ge 1$ and $0<x\le\frac{1}{\sqrt{n}}$. Though the case $n=2$, $x=1$ is not in the FKG regime, it turns out that when loops are assigned one of two colours independently the marginal on loops of either of the colours does satisfy the FKG property. The colouring of loops allows to view the loop $O(2)$ model with $x=1$ as coupling of two percolation configurations. Studying each of them independently and using the XOR operation we establish existence of macroscopic loops, i.e. level lines in the original setting.
(Based on a joint work with H. Duminil-Copin, and I. Manolescu)

ThursdayJan 04, 201813:30
Geometric Functional Analysis and Probability SeminarRoom 290C
Speaker:Sebastien Bubeck (Microsoft Research) and Percy Deift (Courant Institute)Title:Double SeminarAbstract:opens in new windowin html    pdfopens in new window

Sebastien Bubeck (Microsoft Research)

Title: k-server via multiscale entropic regularization
Abstract: I will start by describing how mirror descent is a natural strategy for online decision making, specifically in online learning and metrical task systems. To motivate the k-server problem I will also briefly recall what we know and what we don't know for structured state/action spaces in these models. Using the basic mirror descent calculations I will show how to easily obtain a log(k)-competitive algorithm for k-paging. I will then introduce our new parametrization of fractional k-server on a tree, and explain how to analyze the movement cost of entropy-regularized mirror descent on this parametrization. This leads to a depth*log(k)-competitive (fractional) algorithm for general trees, and log^2(k) for HSTs. I will also briefly mention dynamic embeddings to go beyond the standard log(n) loss in the reduction from general metrics to HSTs.
Joint work with Michael B. Cohen, James R. Lee, Yin Tat Lee, and Aleksander Madry.

Percy Deift (Courant Institute )

Title: Universality in numerical analysis with some examples of cryptographic algorithms.
Abstract: We show that a wide variety of numerical algorithms with random data exhibit universality. Most of the results are computational, but in some important cases universality is established rigorously. We also discuss universality for some cryptographic algorithms.
Joint work with C. Pfrany, G. Menon, S. Olver, T. Trogdan and S. Miller.

ThursdayDec 28, 201713:30
Geometric Functional Analysis and Probability SeminarRoom 290C
Speaker:Amir Dembo (Stanford) and Yuval Peres (Microsoft)Title:DOUBLE TALKAbstract:opens in new windowin html    pdfopens in new window

Talk 1: Amir Dembo (Stanford) 

Title: Large deviations theory for chemical reaction networks.
The microscopic dynamics of well-stirred networks of chemical reactions are modeled as jump Markov processes. At large volume, one may expect in this framework to have a straightforward application of large deviation theory. This is not at all true, for the jump rates are typically neither globally Lipschitz, nor bounded away from zero, with both blowup and absorption as quite possible scenarios. In joint work with Andrea Agazzi and Jean-Pierre Eckman, we utilize Lyapunov stability theory to bypass this challenge and to characterize a large class of network topologies that satisfy the full Wentzell-Freidlin theory of asymptotic rates of exit from domains of attraction.

Talk 2: Yuval Peres (Microsoft)
Title: Trace reconstruction for the deletion channel
Abstract: In the trace reconstruction problem, an unknown string $x$ of $n$ bits is observed through the deletion channel, which deletes each bit with some constant probability q, yielding a contracted string. How many independent outputs (traces) of the deletion channel are needed to reconstruct $x$ with high probability?
The best lower bound known is linear in $n$. Until 2016, the best upper bound was exponential in the square root of $n$. We improve the square root to a cube root using statistics of individual output bits and some inequalities for Littlewood polynomials on the unit circle. This bound is sharp for reconstruction algorithms that only use this statistical information. (Similar results were obtained independently and concurrently by De O'Donnell and Servedio). If the string $x$ is random, we can show a subpolynomial number of traces suffices by comparison to a random walk. (Joint works with Fedor Nazarov, STOC 2017, with Alex Zhai, FOCS 2017 and with Nina Holden & Robin Pemantle, preprint (2017).)

ThursdayDec 21, 201713:30
Geometric Functional Analysis and Probability SeminarRoom 290C
Speaker:Nadav YeshaTitle:CLT for small scale mass distribution of toral Laplace eigenfunctionsAbstract:opens in new windowin html    pdfopens in new window

 In this talk we discuss the fine scale $L^2$-mass distribution of toral Laplace eigenfunctions with respect to random position. For the 2-dimensional torus, under certain flatness assumptions on the Fourier coefficients of the eigenfunctions and generic restrictions on energy levels, both the asymptotic shape of the variance and the limiting Gaussian law are established, in the optimal Planck-scale regime. We also discuss the 3-dimensional case, where the asymptotic behaviour of the variance is analysed in a more restrictive scenario. This is joint work with Igor Wigman.

ThursdayDec 07, 201713:30
Geometric Functional Analysis and Probability SeminarRoom 290C
Speaker:Matan Harel Title:Discontinuity of the phase transition for the planar random-cluster and Potts models with $q > 4$Abstract:opens in new windowin html    pdfopens in new window

The random-cluster model is a dependent percolation model where the weight of a configuration is proportional to q to the power of the number of connected components. It is highly related to the ferromagnetic q-Potts model, where every vertex is assigned one of q colors, and monochromatic neighbors are encouraged. Through non-rigorous means, Baxter showed that the phase transition is first-order whenever $q > 4$ - i.e. there are multiple Gibbs measures at criticality. We provide a rigorous proof of this claim. Like Baxter, our proof uses the correspondence between the above models and the six-vertex model, which we analyze using the Bethe ansatz and transfer matrix techniques. We also prove Baxter's formula for the correlation length of the models at criticality.
This is joint work with Hugo Duminil-Copin, Maxime Gangebin, Ioan Manolescu, and Vincent Tassion.

ThursdayNov 30, 201713:30
Geometric Functional Analysis and Probability SeminarRoom 290C
Speaker:Fanny Augeri Title:Large deviations principles for random matricesAbstract:opens in new windowin html    pdfopens in new windowPlease note that the seminar's starting time has been permanently changed to 13:30

In this talk, I will try to present some techniques to handle the problem of large deviations of the spectrum of random matrices. I will focus on the case of macroscopic statistics of the spectrum of Hermitian matrices - in particular Wigner matrices - as the empirical distribution of the eigenvalues, the largest eigenvalue or the traces of powers.

In a first part, I will be concerned with the so-called "objective method''. Coined by David Aldous, this method consists in introducing, given a sequence of random objects, like random finite graphs, a new infinite random object from which one can deduce asymptotic properties of the original sequence. In the context of random matrices, this method has been mainly advertised by Balint Virag, and proven effective in showing universality results for the so-called beta-ensembles. Regarding large deviations of random matrices, this "objective method'' amounts to embed our sequence of matrices with growing size into an appropriate space on which one is able to understand the large deviations, and carry out a contraction principle. I will review several large deviations principles obtained by this method, given by interpretations of random matrices as either dense or sparse graphs, and point out the limits of this strategy.

ThursdayNov 23, 201714:10
Geometric Functional Analysis and Probability SeminarRoom 290C
Speaker:Naomi Feldheim Title:Persistence of Gaussian Stationary ProcessesAbstract:opens in new windowin html    pdfopens in new window

Consider a real Gaussian stationary process, either on Z or on R.
What is the probability that it remains positive on [0,N] for large N?

The relation between this probability, known as the persistence probability, and the covariance kernel of the process has been investigated since the 1950s with motivations stemming from probability, engineering and mathematical physics. Nonetheless, until recently, good estimates were known only for particular cases, or when the covariance kernel is either non-negative or summable.

In the first hour of the talk we will discuss new spectral methods which greatly simplify the analysis of persistence. We will then describe its qualitative behavior in a very general setting.

In the second hour, we will describe (very) recent progress. In particular we will show the proof of the "spectral gap conjecture'', which states: if the spectral measure vanishes on an interval containing 0 then the persistence is less then e^{-cN^2}, and this bound is tight if the measure is non-singular and compactly supported. 
Time permitting, we will also discuss "tiny persistence'' phenomena (of the order of e^{-e^{cN}}).

Based on joint works with Ohad Feldheim, Benjamin Jaye, Fedor Nazarov and Shahaf Nitzan.

ThursdayNov 09, 201714:00
Geometric Functional Analysis and Probability SeminarRoom 290C
Speaker:Ilya GoldsheidTitle:Real and complex eigenvalues of the non-self-adjoint Anderson model.Abstract:opens in new windowin html    pdfopens in new windowNOTE UNUSUAL TIME
TBA
ThursdayJun 29, 201711:15
Geometric Functional Analysis and Probability SeminarRoom 290C
Speaker:Amir Dembo Title:The criticality of a randomly-driven front.Abstract:opens in new windowin html    pdfopens in new window
Consider independent continuous-time random walks on the integers to the right of a front R(t). Starting at R(0)=0, whenever a particle attempts to jump into the front, the latter instantaneously advances k steps to the right, absorbing all particles along its path. Sly (2016) resolves the question of Kesten and Sidoravicius (2008), by showing that for k=1 the front R(t) advances linearly once the particle density exceeds 1, but little is known about the large t asymptotic of R(t) at critical density 1. In a joint work with L-C Tsai, for the variant model with k taken as the minimal random integer such that exactly k particles are absorbed by the move of R(t), we obtain both scaling exponent and the random scaling limit for the front at the critical density 1. Our result unveils a rarely seen phenomenon where the macroscopic scaling exponent is sensitive to the initial local fluctuations (with the scaling limit oscillating between instantaneous super and sub-critical phases).
ThursdayJun 15, 201711:15
Geometric Functional Analysis and Probability SeminarRoom 290C
Speaker:Piotr Nayar Title:Gaussian mixtures with applications to entropy inequalities and convex geometryAbstract:opens in new windowin html    pdfopens in new window
We say that a symmetric random variable X is a Gaussian mixture if X has the same distribution as YG, where G is a standard Gaussian random variable, and Y is a positive random variable independent of G. In the first part of the talk we use this simple notion to study the Shannon entropy of sums of independent random variables. In the second part we investigate, using Gaussian mixtures, certain topics related to the geometry of B_p^n balls, including optimal Khinchine-type inequalities and Schur-type comparison for volumes of section and projections of these sets. In the third part we discuss extensions of Gaussian correlation inequality to the case of p-stable laws and uniform measure on the Euclidean sphere. Based on a joint work with Alexandros Eskenazis and Tomasz Tkocz.
ThursdayJun 08, 201711:15
Geometric Functional Analysis and Probability SeminarRoom 290C
Speaker:Nishant ChandgotiaTitle:Irrational rotations, random affine transformations and the central limit theoremAbstract:opens in new windowin html    pdfopens in new window
It is a well-known result from Hermann Weyl that if alpha is an irrational number in [0,1) then the number of visits of successive multiples of alpha modulo one in an interval contained in [0,1) is proportional to the size of the interval. In this talk we will revisit this problem, now looking at finer joint asymptotics of visits to several intervals with rational end points. We observe that the visit distribution can be modelled using random affine transformations; in the case when the irrational is quadratic we obtain a central limit theorem as well. Not much background in probability will be assumed. This is in joint work with Jon Aaronson and Michael Bromberg.
ThursdayMar 09, 201711:15
Geometric Functional Analysis and Probability SeminarRoom 290C
Speaker:Sasha ShamovTitle:Conditional determinantal processes are determinantalAbstract:opens in new windowin html    pdfopens in new window

A determinantal point process governed by a locally trace class Hermitian contraction kernel on a measure space $E$ remains determinantal when conditioned on its configuration on an arbitrary measurable subset $B \subset E$. Moreover, the conditional kernel can be chosen canonically in a way that is "local" in a non-commutative sense, i.e. invariant under "restriction" to closed subspaces $L^2(B) \subset P \subset L^2(E)$.

Using the properties of the canonical conditional kernel we establish a conjecture of Lyons and Peres: if $K$ is a projection then almost surely all functions in its image can be recovered by sampling at the points of the process.

ThursdayFeb 09, 201711:15
Geometric Functional Analysis and Probability SeminarRoom 290C
Speaker:Alexander FishTitle:The values of quadratic forms on difference sets, measure rigidity and equidistributionAbstract:opens in new windowin html    pdfopens in new window

Given a quadratic form Q in d variables over the integers, e.g. Q(x,y,z) = xy - z^2, and a set of positive density E in Z^d, we investigate what kind of structure can be found in the set Q(E-E). 
We will see that if d >= 3, and Q is indefinite, then the measure rigidity, due to Bourgain-Furman-Lindenstrauss-Mozes or Benoist-Quint, of the action of the group of the symmetries of Q implies that there exists k >=1 such that  k^2*Q(Z^d) is a subset of Q(E-E). 
We will give an alternative proof of the theorem for the case Q(x,y,z) = xy - z^2 that uses more classical equidistribution results of Vinogradov, and Weyl, as well as a more recent result by Frantzikinakis-Kra. The latter proof extends the theorem to other polynomials having a much smaller group of symmetries. Based on joint works with M. Bjorklund (Chalmers), and K. Bulinski (Sydney). 

ThursdayJan 19, 201711:15
Geometric Functional Analysis and Probability SeminarRoom 290C
Speaker:Jay Rosen Title:Tightness for the Cover Time of S^2 Abstract:opens in new windowin html    pdfopens in new window

Let M be a smooth, compact, connected two-dimensional, Riemannian manifold without boundary, and let  C_epsilon be  the amount of time needed for the Brownian motion to come within (Riemannian) distance epsilon of all points in M. The first order asymptotics of C_epsilon as epsilon goes to 0 are known. We show that for the two dimensional sphere 

\sqrt{C_epsilon}-2\sqrt{2}\( \log \epsilon^{-1}- \frac{1}{4}\log\log \epsilon^{-1}\) is tight.

Joint work with David Belius and  Ofer Zeitouni.

ThursdayJan 12, 201711:00
Geometric Functional Analysis and Probability SeminarRoom 290C
Speaker:Ran Tessler and Assaf Naor Title:Double lecture !Abstract:opens in new windowin html    pdfopens in new window

First Speaker: Ran Tessler (ETH)
Time: 11:00
Title: A sharp threshold for Hamiltonian spheres in a random 2-complex.
Abstract: We define the notion of Hamiltonian sphere - a 2-complex homeomorphic to a sphere which uses all vertices. We prove an explicit sharp threshold for the appearance of Hamiltonian spheres in the Linial-Meshulam model for random 2-complexes. The proof combines combinatorial, probabilistic and geometric arguments. Based on a joint work with Zur Luria.

Second Speaker: Assaf Naor (Princeton)
Time: 12:00
Title: A new vertical-versus-horizontal isoperimetric inequality on the Heisenberg group, with applications to metric geometry and approximation algorithms
Abstract: In this talk we will show that for every measurable subset of the Heisenberg group of dimension at least 5, an appropriately defined notion of its "vertical perimeter" is at most a constant multiple of its horizontal (Heisenberg) perimeter. We will explain how this new isoperimetric-type inequality solves open questions in analysis (an endpoint estimate for a certain singular integral on W^{1,1}), metric geometry (sharp nonembeddability into L_1) and approximation algorithms (asymptotic evaluation of the performance of the Goemans-Linial algorithm for the Sparsest Cut problem). Joint work with Robert Young.

ThursdayJan 05, 201711:00
Geometric Functional Analysis and Probability SeminarRoom 290C
Speaker:Amir DemboTitle:Walking within growing domains: recurrence versus transience Abstract:opens in new windowin html    pdfopens in new window
When is simple random walk on growing in time d-dimensional domains recurrent? For domain growth which is independent of the walk, we review recent progress and related universality conjectures about a sharp recurrence versus transience criterion in terms of the growth rate. We compare this with the question of recurrence/transience for time varying conductance models, where Gaussian heat kernel estimates and evolving sets play an important role. We also briefly contrast such expected universality with examples of the rich behavior encountered when monotone interaction enforces the growth as a result of visits by the walk to the current domain's boundary. This talk is based on joint works with Ruojun Huang, Ben Morris, Yuval Peres, Vladas Sidoravicius and Tianyi Zheng.
ThursdayDec 29, 201611:00
Geometric Functional Analysis and Probability SeminarRoom 290C
Speaker:Alon Nishry Title:Gaussian complex zeros on the hole event: the emergence of a forbidden regionAbstract:opens in new windowin html    pdfopens in new window

Consider the Gaussian Entire Function (GEF) whose Taylor coefficients are independent complex-valued Gaussian variables, and the variance of the k-th coefficient is 1/k!. This random Taylor series is distinguished by the invariance of its zero set with respect to the isometries of the complex plane.
I will show that the law of the zero set, conditioned on the GEF having no zeros in a disk of radius r, and properly normalized, converges to an explicit limiting Radon measure in the plane, as r goes to infinity. A remarkable feature of this limiting measure is the existence of a large 'forbidden region' between a singular part supported on the boundary of the (scaled) hole and the equilibrium measure far from the hole. This answers a question posed by Nazarov and Sodin, and is in stark contrast to the corresponding result known to hold in the random matrix setting, where such a gap does not appear.
The talk is based on a joint work with S. Ghosh.

ThursdayDec 15, 201611:00
Geometric Functional Analysis and Probability SeminarRoom 290C
Speaker:Snir Ben OvadiaTitle:Symbolic dynamics for non uniformly hyperbolic diffeomorphisms of compact smooth manifolds Abstract:opens in new windowin html    pdfopens in new window

Given a dynamical system, a partition of the space induces a mapping to the space of sequences of the partition elements (a point is mapped to the partition elements containing its orbit terms). Such a duality is called Symbolic Dynamics, Markov partitions are an important tool, as the symbolic dynamics they induce enfold many of the important dynamical properties of the original system, and they allow an easier studying of them.
We show that general non uniformly hyperbolic C^{1+epsilon} diffeomorphism on compact manifolds of any dimension admit countable Markov partitions. Previously this was only known in dimension 2.

ThursdayNov 17, 201611:00
Geometric Functional Analysis and Probability SeminarRoom 290C
Speaker:Anirban Basak Title:Invertibility of sparse random matricesAbstract:opens in new windowin html    pdfopens in new window
We consider a class of sparse random matrices of the form $A_n =(\xi_{i,j}\delta_{i,j})_{i,j=1}^n$, where $\{\xi_{i,j}\}$ are i.i.d.~centered random variables, and $\{\delta_{i,j}\}$ are i.i.d.~Bernoulli random variables taking value $1$ with probability $p_n$, and prove a quantitative estimate on the smallest singular value for $p_n = \Omega(\frac{\log n}{n})$, under a suitable assumption on the spectral norm of the matrices. This establishes the invertibility of a large class of sparse matrices. We also find quantitative estimates on the smallest singular value of the adjacency matrix of a directed Erdos-Reyni graph whenever its edge connectivity probability is above the critical threshold $\Omega(\frac{\log n}{n})$. This is joint work with Mark Rudelson.
ThursdayNov 03, 201611:00
Geometric Functional Analysis and Probability SeminarRoom 290C
Speaker:David Ellis Title:Some applications of the $p$-biased measure to Erd\H{o}s-Ko-Rado type problemsAbstract:opens in new windowin html    pdfopens in new window

If $X$ is a finite set, the $p$-biased measure on the power-set of $X$ is defined as follows: choose a subset $S$ of $X$ at random by including each element of $X$ independently with probability $p$. If $\mathcal{F}$ is a family of subsets of $X$, one can consider the {\em $p$-biased measure} of $\mathcal{F}$, denoted by $\mu_p(\mathcal{F})$, as a function of $p$; if $\mathcal{F}$ is closed under taking supersets, then this function is an increasing function of $p$. Seminal results of Friedgut and Friedgut-Kalai give criteria for this function to have a 'sharp threshold'. A careful analysis of the behaviour of this function also yields some rather strong results in extremal combinatorics which do not explicitly mention the $p$-biased measure - in particular, in the field of {\em Erd\H{o}s-Ko-Rado type problems}, which concern the sizes of families of objects in which any two (or three...) of the objects 'agree' or 'intersect' in some way. We will discuss some of these, including a recent proof of an old conjecture of Frankl that a symmetric three-wise intersecting family of subsets of $\{1,2,\ldots,n\}$ has size $o(2^n)$, and some 'stability' results characterizing the structure of 'large' $t$-intersecting families of $k$-element subsets of $\{1,2,\ldots,n\}$. Based on joint work with (subsets of) Nathan Keller, Noam Lifshitz and Bhargav Narayanan.

WednesdaySep 14, 201614:00
Geometric Functional Analysis and Probability SeminarRoom 290C
Speaker:Assaf NaorTitle:The Lipschitz extension problem for finite dimensional normed spacesAbstract:opens in new windowin html    pdfopens in new windowPLEASE NOTE UNUSUAL DAY
ThursdayJun 23, 201612:00
Geometric Functional Analysis and Probability SeminarRoom 290C
Speaker:Shamgar Gurevich Title:Small Representations of Finite Classical GroupsAbstract:opens in new windowin html    pdfopens in new window
Many properties of a finite group G can be approached using formulas involving sums over its characters. A serious obstacle in applying these formulas seemed to be lack of knowledge over the low dimensional representations of G. In fact, the "small" representations tend to contribute the largest terms to these sums, so a systematic knowledge of them might lead to proofs of some conjectures which are currently out of reach.
ThursdayJun 23, 201611:15
Geometric Functional Analysis and Probability SeminarRoom 290C
Speaker:Jonathan HermonTitle:L_2 Mixing and hypercontractivity via maximal inequalities and hitting-timesAbstract:opens in new windowin html    pdfopens in new window

There are numerous essentially equivalent characterizations of mixing in $L_1$ of a finite Markov chain. Some of these characterizations involve natural probabilistic concepts such as couplings, stopping times and hitting times. In contrast, while there are several analytic and geometric tools for bounding the $L_2$ mixing time, none of them are tight and they do not have a probabilistic interpretation.

We provide tight probabilistic characterizations in terms of hitting times distributions for mixing in $L_2$ (for normal chains) and (under reversibility) in relative entropy. This is done by assigning appropriate penalty (depending on the size of the set) to the case that the chain did not escape from a certain set.

We also prove a new extremal characterization of the log-sobolev constant in terms of a weighted version of the spectral gap (where the weight depends on the size of the support of the function).

ThursdayJun 16, 201612:00
Geometric Functional Analysis and Probability SeminarRoom 290C
Speaker:Eliran SubagTitle:Critical points and the Gibbs measure of pure spherical spin glassesAbstract:opens in new windowin html    pdfopens in new window
Recently, several results concerning the critical points of the energy landscape of pure $p$-spin spherical spin glasses have been obtained by means of moment computations and a proof of a certain invariance property. I will describe those and explain how they can be boosted by an investigation of the behavior around the critical points to obtain a geometric description for the Gibbs measure at low enough temperature. The talk is based on joint work with Ofer Zeitouni.
ThursdayJun 16, 201611:15
Geometric Functional Analysis and Probability SeminarRoom 290C
Speaker:Eviatar ProcacciaTitle:Can one hear the shape of a random walk?Abstract:opens in new windowin html    pdfopens in new window
We consider a Gibbs distribution over random walk paths on the square lattice, proportional to a random weight of the path's boundary. We show that in the zero temperature limit, the paths condensate around an asymptotic shape. This limit shape is characterized as the minimizer of the functional, mapping open connected subsets of the plane to the sum of their principle eigenvalue and perimeter (with respect to the first passage percolation norm). A prime novel feature of this limit shape is that it is not in the class of Wulff shapes. Joint work with Marek Biskup.
ThursdayMay 05, 201611:00
Geometric Functional Analysis and Probability Seminar
Speaker:Ilya GoldsheidTitle:Recurrent Random Walks on a Strip: conditions for the CLTAbstract:opens in new windowin html    pdfopens in new window Double feature room 155
This is joint work with Dima Dolgopyat. We prove that a recurrent random walk (RW) in i.i.d. random environment (RE) on a strip which does not obey the Sinai law exhibits the Central Limit asymptotic behaviour. Moreover, there exists a collection of proper subvarieties in the space of transition probabilities such that: (a) If the RE is stationary and ergodic and the transition probabilities are concentrated on one of sub-varieties from our collection then the CLT holds; (b) If the environment is i.i.d then the above condition is also necessary for the CLT to hold. In particular, the CLT holds for the quasiperiodic environments with Diophantine frequencies in the recurrent case and complement this result by proving that in the transient case the CLT holds for all strictly ergodic environments.
ThursdayMay 05, 201611:00
Geometric Functional Analysis and Probability Seminar
Speaker:Tal OrenshteinTitle:One-dependent walks in hypergeometric-Dirichlet environmentsAbstract:opens in new windowin html    pdfopens in new windowDouble feature room 155
Dirichlet environments are one of the few examples in Random Walk in Random Environment in which some non-trivial random walk properties are fully and explicitly characterized in terms of the parameters. A key feature of the model is the so-called 'time reversal property', saying that inverting the time is resulting in the same class of models, with an explicit change of parameters. In this talk, which is based on a joint work in process with Christophe Sabot, I'll present a generalization of random walks in Dirichlet environments using hypergeometric functions having that nice feature, and discuss the question of existence of an invariant probability measure for the process on the environments from the point of view of the walker which is absolutely continuous with respect to the initial measure.
ThursdayApr 21, 201611:00
Geometric Functional Analysis and Probability Seminar
Speaker:Atilla YilmazTitle:Large deviations for random walk in space-time random environment: averaged vs. quenchedAbstract:opens in new windowin html    pdfopens in new windowRoom 155
I will present recent joint work with F. Rassoul-Agha (Utah) and T. Seppalainen (Madison) where we consider random walk on a hypercubic lattice of arbitrary dimension in a space-time random environment that is assumed to be temporally independent and spatially translation invariant. The large deviation principle (LDP) for the empirical velocity of the averaged walk (i.e., level-1) is simply Cramer’s theorem. We take the point of view of the particle and establish the process-level (i.e., level-3) averaged LDP for the environment Markov chain. The rate function $I_{3,a}$ is a specific relative entropy which reproduces Cramer’s rate function via the so-called contraction principle. We identify the unique minimizer of this contraction at any velocity and analyse its structure. When the environment is spatially ergodic, the level-3 quenched LDP follows from our previous work which gives a variational formula for the rate function $I_{3,q}$ involving a Donsker-Varadhan-type relative entropy $H_q$. We derive a decomposition formula for $I_{3,a}$ that expresses it as a sum of contributions from the walk (via $H_q$) and the environment. We use this formula to characterize the equality of the level-1 averaged and quenched rate functions, and conclude with several related results and open problems.
ThursdayMar 31, 201611:05
Geometric Functional Analysis and Probability SeminarRoom 261
Speaker:Amir Yehudayoff Title:Geometric stability using information theoryAbstract:opens in new windowin html    pdfopens in new windowmoved to room 155

Projection inequalities bound the volume of a body in terms of a product of the volumes of lower-dimensional projections of the body. Two well-known examples are the Loomis-Whitney inequality and the more general Uniform Cover inequality. We will see how to use information theory to prove stability versions of these inequalities, showing that when they are close to being tight, the body in question is close to being a box (which is the unique case of equality). We will also see how to obtain a stability result for the edge-isoperimetric inequality in the infinite d-dimensional lattice. Namely, that a subset of Z^d with small edge-boundary must be close in symmetric difference to a d-dimensional cube.

Based on work with David Ellis, Ehud Friedgut and Guy Kindler.

ThursdayMar 17, 201611:05
Geometric Functional Analysis and Probability SeminarRoom 261
Speaker:Asaf FerberTitle:Iterated Log Law for various graph parametersAbstract:opens in new windowin html    pdfopens in new window

We show that a version of the classical Iterated Log Law of Khinchin, and independently of Kolmogorov from the 1920's, holds for various parameters in the binomial random graph model and in a random 0/1 Bernoulli matrix. In particular, for a constant p, we show that such a law holds for the number of copies of a fixed graph H in G(n,p), we show a similar statement for the number of Hamilton cycles in a random k-uniform hypergraph, provided that k\geq 4. In the graph case (that is, k=2), since the number of Hamilton cycles in G(n,p), denoted by X_n, does not converge to a normal distribution but rather tends to a log-normal distribution (as has been first proved by Janson), we show that a version of the Iterated Log Law holds for \log X_n. We also obtain similar result for the permanent of a 0/1 bernouli random matrix.

No prior knowledge is required.

Joint with Daniel Motealegre and Van Vu.

ThursdayMar 10, 201611:05
Geometric Functional Analysis and Probability SeminarRoom 261
Speaker:Mark RudelsonTitle:No-gaps delocalization for general random matricesAbstract:opens in new windowin html    pdfopens in new window

Heuristically, delocalization for a random matrix means that its normalized eigenvectors look like the vectors uniformly distributed over the unit sphere. This can be made precise in a number of different ways. We show that with high probability, any sufficiently large set of coordinates of an eigenvector carries a non-negligible portion of its Euclidean norm. Our results pertain to a large class of random matrices including matrices with independent entries, symmetric, skew-symmetric matrices, as well as more general ensembles.

Joint work with Roman Vershynin.

ThursdayMar 03, 201611:05
Geometric Functional Analysis and Probability SeminarRoom 261
Speaker:Alexei V. PenskoiTitle:Recent advances in geometric optimization of eigenvalues of the Laplace-Beltrami operator on closed surfacesAbstract:opens in new windowin html    pdfopens in new window
Since a metric defines the Laplace-Beltrami operator on a closed surface, the eigenvalues of the Laplace-Beltrami operator are functionals on the space of Riemannian metrics on the surface. A metric is called maximal for i-th eigenvalue if the i-th eigenvalue attends its maximum on this metric. It turns out that the question about finding maximal metrics is very deep and related to analysis, topology, algebraic and differential geometry. In this talk several recent advances in this question will be exposed.
ThursdayFeb 18, 201611:15
Geometric Functional Analysis and Probability SeminarRoom 261
Speaker:Evgeny StrahovTitle:Determinantal processes related to products of random matricesAbstract:opens in new windowin html    pdfopens in new window
I will talk about determinantal processes formed by eigenvalues and singular values of products of complex Gaussian matrices. Such determinantal processes can be understood as natural generalizations of the classical Ginibre and Laguerre ensembles of Random Matrix Theory, and the correlation kernels of these processes can be expressed in terms of special functions/double contour integrals. This enables to investigate determinantal processes for products of random matrices in different asymptotic regimes, and to compute different probabilistic quantities of interest. In particular, I will present the asymptotics for the hole probabilities, i.e. for probabilities of the events that there are no particles in a disc of radius r with its center at 0, as r goes to infinity. In addition, I will explain how the gap probabilities for squared singular values of products of random complex matrices can be described in terms of completely integrable Hamiltonian differential equations, and how to interpret these Hamiltonian differential equations as the monodromy preserving deformation equations of the Jimbo, Miwa, Mori, Ueno and Sato theory. Finally, I will discuss certain time-dependent determinantal processes related to products of random matrices.
ThursdayFeb 04, 201611:15
Geometric Functional Analysis and Probability SeminarRoom 261
Speaker:Gaultier Lambert Title:Fluctuations of linear statistics of determinantal processesAbstract:opens in new windowin html    pdfopens in new window
Determinantal point processes arise in the description of eigenvalues of unitary invariant Hermitian random matrices, as well as in many statistical mechanics models such as random tilings, non-intersecting paths, etc. I will explain a cumulant method developed by A. Soshnikov to analyze the asymptotics distributions of linear statistics of determinantal processes and certain combinatorial identities associated with the sine process. I will present some applications to orthogonal ensembles and, if time permits, to certain biorthogonal ensembles and discuss some models which exhibit a transition from Poisson to GUE.
ThursdayJan 28, 201611:05
Geometric Functional Analysis and Probability SeminarRoom 261
Speaker:Nathan KellerTitle:Stability Versions of Erdös-Ko-Rado Type Theorems via Isoperimetry Abstract:opens in new windowin html    pdfopens in new window

Erdös-Ko-Rado (EKR) type theorems yield upper bounds on the size of set families under various intersection requirements on the elements. Stability versions of such theorems assert that if the size of a family is close to the maximum possible then the family itself must be close (in appropriate sense) to a maximum family. In this talk we present an approach to stability versions of EKR-type theorems through isoperimetric inequalities for subsets of the hypercube. We use this approach to obtain tight stability versions of the EKR theorem itself and of the Ahlswede-Khachatrian theorem on t-intersecting families (for k < n/(t+1)), and to show that, somewhat surprisingly, both theorems hold when the "intersection" requirement is replaced by a much weaker requirement. Furthermore, we obtain stability versions of several recent EKR-type results, including Frankl's proof of the Erdös matching conjecture for n>(2s+1)k-s.

Joint work with David Ellis and Noam Lifshitz.

ThursdayJan 07, 201611:05
Geometric Functional Analysis and Probability SeminarRoom 261
Speaker:Shamgar GurevitchTitle:Low Dimensional Representations of Finite Classical GroupsAbstract:opens in new windowin html    pdfopens in new window

Many questions about properties of a finite group such as random walks, spectrum of Cayley graphs, distribution of word maps etc., can be approached by using “generalized Fourier sum” formulas involving characters of the group. Numerical data show that characters of low dimensional representations of the group contribute the largest terms to these sums. However, relatively little seems to be known about these small representations so a systematic knowledge of them could lead to proofs of some of the properties. The talk will demonstrates, through concrete examples, and numerical simulations, a new method to construct and analyze those small representations, and hence hopefully to solve some of the aforementioned questions.

The talk is intended for non-experts.

This is part from a joint project with Roger Howe (Yale).

WednesdayJan 06, 201611:05
Geometric Functional Analysis and Probability SeminarRoom 261
Speaker:Eyal Lubetzky Title:Effect of initial conditions on mixing for spin systemsAbstract:opens in new windowin html    pdfopens in new windownote unusual day

Recently, the "information percolation" framework was introduced as a way to obtain sharp estimates on mixing for spin systems at high temperatures, and in particular, to establish cutoff for the Ising model in three dimensions up to criticality from a worst starting state. I will describe how this method can be used to understand the effect of different initial states on the mixing time, both random (''warm start'') and deterministic.

Joint work with Allan Sly.

ThursdayDec 31, 201511:05
Geometric Functional Analysis and Probability SeminarRoom 261
Speaker:Zemer KosloffTitle:Symmetric Birkhoff sums in infinite ergodic theoryAbstract:opens in new windowin html    pdfopens in new window

By a Theorem of Aaronson, normalized Birkhoff sums of positive integrable functions in infinite, ergodic systems either tend to 0 almost surely or there is a subsequence along which every further subsequence tends to infinity. This is not true for normalized symmetric Birkhoff sums where the summation is along a symmetric time interval as there are examples of infinite, ergodic systems for which the absolutely normalized symmetric Birkhoff sums of positive integrable functions may be almost surely bounded away from zero and infinity. In this talk I will start by explaining a variety of transformations (of different nature) satisfying this phenomena, discuss the case main result that the absolutely normalized, symmetric Birkhoff sums of positive integrable functions in infinite, ergodic systems never converge point-wise and there even exists a universal divergence statement. Time permits I will show some examples of actions of other groups which converge and some recent (yet unwritten) results on actions by commuting skew products which are related to self intersection local times.

The contents of this talk are a combination of 3 papers, one of which is a joint work with Benjamin Weiss and Jon Aaronson and another one is work in progress with Jon Aaronson.

ThursdayDec 17, 201511:00
Geometric Functional Analysis and Probability SeminarRoom 261
Speaker:Amir DemboTitle:Extremal Cuts of Sparse Random GraphsAbstract:opens in new windowin html    pdfopens in new window

The Max-Cut problem seeks to determine the maximal cut size in a given graph. With no polynomial-time efficient approximation for Max-Cut (unless P=NP), its asymptotic for a typical large sparse graph is of considerable interest. We prove that for uniformly random d-regular graph of N vertices, and for the uniformly chosen Erdos-Renyi graph of M=Nd/2 edges, the leading correction to M/2 (the typical cut size), is P∗sqrt(NM/2). Here P∗ is the ground state energy of the Sherrington-Kirkpatrick model, expressed analytically via Parisi's formula.

This talk is based on a joint work with Subhabrata Sen and Andrea Montanari.

ThursdayDec 10, 201511:00
Geometric Functional Analysis and Probability SeminarRoom 261
Speaker:Ohad FeldheimTitle:Double Roots of Random PolynomialsAbstract:opens in new windowin html    pdfopens in new window

We consider random polynomials of degree n whose coefficients are i.i.d. distributed over a finite set of integers, with probability at most 1/2 to take any particular value. We show that the probability that such a polynomial of degree n has a double root is dominated by the probability that 0,1 or -1 are double roots up to an error of o(n−2). Our result generalizes a similar result of Peled, Sen and Zeitouni for Littlewood polynomials.

Joint work with Ron Peled and Arnab Sen.

ThursdayDec 03, 201511:00
Geometric Functional Analysis and Probability SeminarRoom 261
Speaker:Ron RosenthalTitle:Eigenvalue confinement and spectral gap for random simplicial complexesAbstract:opens in new windowin html    pdfopens in new window
We consider the adjacency operator of the Linial-Meshulam model for random simplicial complexes on $n$ vertices, where each $d$-cell is added independently with probability $p$ to the complete $(d-1)$-skeleton. From the point of view of random matrix theory, the adjacency matrix is a sparse, self adjoint random matrix with dependent entries. Under the assumption $np(1-p)>> log^4 n$, we prove that the spectral gap between the $\binom{n-1}{d}$ smallest eigenvalues and the remaining $\binom{n-1}{d-1}$ eigenvalues is $np-2\sqrt{dnp(1-p)}(1+o(1))$ with high probability. This estimate follows from a more general result on eigenvalue confinement. In addition, we prove that the global distribution of the eigenvalues is asymptotically given by the semicircle law. Based on a joint work with Antti Knowles.
ThursdayNov 26, 201511:00
Geometric Functional Analysis and Probability SeminarRoom 261
Speaker:Yaar SolomonTitle:The Danzer problem and a solution to a problem of Gowers Abstract:opens in new windowin html    pdfopens in new window
Is there a point set Y in R^d, and C>0, such that every convex set of volume 1 contains at least one point of Y and at most C? This discrete geometry problem was posed by Gowers in 2000, and it is a special case of an open problem posed by Danzer in 1965. I will present two proofs that answers Gowers' question with a NO. The first approach is dynamical; we introduce a dynamical system and classify its minimal subsystems. This classification in particular yields the negative answer to Gowers' question. The second proof is direct and it has nice applications in combinatorics. The talk will be accessible to a general audience. [This is a joint work with Omri Solan and Barak Weiss].
ThursdayNov 12, 201511:00
Geometric Functional Analysis and Probability SeminarRoom 261
Speaker:Christopher JoynerTitle:Random Walk approach to spectral statistics in random Bernoulli matricesAbstract:opens in new windowin html    pdfopens in new window
Random Bernoulli matrices (in which the matrix elements are chosen independently from plus or minus 1 with equal probability) are intimately connected to the adjacency matrices of random graphs and share many spectral properties. In the limit of large matrix dimension the distribution of eigenvalues from such matrices resembles that from matrices in which the elements are chosen randomly from a Gaussian distribution - the question is why? We take a dynamical approach to this problem, which is achieved by initiating a discrete random walk process over the space of matrices. Previously we have used this idea to analyse the corresponding eigenvalue motion but I will discuss some recent developments which involve the adaptation of Stein's method to this context.
ThursdayAug 06, 201511:05
Geometric Functional Analysis and Probability SeminarRoom 208
Speaker:Balázs RáthTitle:Voter model percolationAbstract:opens in new windowin html    pdfopens in new windowplease note unusual room

The voter model on $\Z^d$ is a particle system that serves as a rough model for changes of opinions among social agents or, alternatively, competition between biological species occupying space. When $d \geq 3$, the set of (extremal) stationary distributions is a family of measures $\mu_\alpha$, for $\alpha$ between 0 and 1. A configuration sampled from $\mu_\alpha$ is a strongly correlated field of 0's and 1's on $\Z^d$ in which the density of 1's is $\alpha$.

We consider such a configuration as a site percolation model on $\Z^d$. We prove that if $d \geq 5$, the probability of existence of an infinite percolation cluster of 1's exhibits a phase transition in $\alpha$. If the voter model is allowed to have sufficiently spread-out interactions, we prove the same result for $d \geq 3$.

These results partially settle a conjecture of Bricmont, Lebowitz and Maes (1987).
Joint work with Daniel Valesin (University of Groningen)

ThursdayJul 30, 201511:05
Geometric Functional Analysis and Probability SeminarRoom 261
Speaker:Eviatar ProcacciaTitle:Stationary Eden model on groupsAbstract:opens in new windowin html    pdfopens in new window

We consider two stationary versions of the Eden model, on the upper half planar lattice, resulting in an infinite forest covering the half plane. Under weak assumptions on the weight distribution and by relying on ergodic theorems, we prove that almost surely all trees are finite. Using the mass transport principle, we generalize the result to Eden model in graphs of the form $G \times Z$, where G is a Cayley graph. This generalizes certain known results on the two-type Richardson model, in particular of Deijfen and Häggström in 2007.

ThursdayJul 16, 201511:05
Geometric Functional Analysis and Probability SeminarRoom 1
Speaker:Alexander FishTitle:Ergodic theorems for amenable groupsAbstract:opens in new windowin html    pdfopens in new windowNOTE UNUSUAL ROOM

We will talk on the validity of the mean ergodic theorem along left Følner sequences in a countable amenable group G. Although the weak ergodic theorem always holds along any left Følner sequence in G, we will provide examples where the mean ergodic theorem fails in quite dramatic ways. On the other hand, if G does not admit any ICC quotients, e.g. if G is virtually nilpotent, then we will prove that the mean ergodic theorem does indeed hold along any left Følner sequence. Based on the joint work with M. Björklund (Chalmers).

ThursdayJul 09, 201511:05
Geometric Functional Analysis and Probability SeminarRoom 261
Speaker:Dan FlorentinTitle:Stability and Rate of Convergence of the Steiner SymmetrizationAbstract:opens in new windowin html    pdfopens in new window
We present a direct analytic method towards an estimate for the rate of convergence (to the Euclidean Ball) of Steiner symmetrizations. To this end we present a modified version of a known stability property of Steiner symmetrization.
ThursdayJul 02, 201511:05
Geometric Functional Analysis and Probability SeminarRoom 261
Speaker:Dan FlorentinTitle:Stability and Rate of Convergence of the Steiner SymmetrizationAbstract:opens in new windowin html    pdfopens in new window
We present a direct analytic method towards an estimate for the rate of convergence (to the Euclidean Ball) of Steiner symmetrizations. To this end we present a modified version of a known stability property of Steiner symmetrization.
ThursdayJun 25, 201511:05
Geometric Functional Analysis and Probability SeminarRoom 261
Speaker:Yinon SpinkaTitle:Long-range order in random 3-colorings of Z^dAbstract:opens in new windowin html    pdfopens in new window

Consider a random coloring of a bounded domain in Zd with the probability of each coloring F proportional to exp(−β∗N(F)), where β>0 is a parameter (representing the inverse temperature) and N(F) is the number of nearest neighboring pairs colored by the same color. This is the anti-ferromagnetic 3-state Potts model of statistical physics, used to describe magnetic interactions in a spin system. The Kotecký conjecture is that in such a model, for d≥3 and high enough β, a sampled coloring will typically exhibit long-range order, placing the same color at most of either the even or odd vertices of the domain. We give the first rigorous proof of this fact for large d. This extends previous works of Peled and of Galvin, Kahn, Randall and Sorkin, who treated the case β=∞.

The main ingredient in our proof is a new structure theorem for 3-colorings which characterizes the ways in which different "phases" may interact, putting special emphasis on the role of edges connecting vertices of the same color. We also discuss several related conjectures. No background in statistical physics will be assumed and all terms will be explained thoroughly.

Joint work with Ohad Feldheim.

ThursdayJun 18, 201511:05
Geometric Functional Analysis and Probability SeminarRoom 261
Speaker:Amir DemboTitle:The Atlas model, in and out of equilibriumAbstract:opens in new windowin html    pdfopens in new window

Consider a one-dimensional semi-infinite system of Brownian particles, starting at Poisson (L) point process on the positive half-line, with the left-most (Atlas) particle endowed a unit drift to the right. We show that for the equilibrium density (L=2), the asymptotic Gaussian space-time particle fluctuations are governed by the stochastic heat equation with Neumann boundary condition at zero. As a by product we resolve a conjecture of Pal and Pitman (2008) about the asympotic (random) fBM trajectory of the Atlas particle.

In a complementary work, we derive and explicitly solve the Stefan (free-boundary) equations for the limiting particle-profile when starting at out of equilibrium density (L other than 2). We thus determine the corresponding (non-random) asymptotic trajectory of the Atlas particle.

This talk is based on joint works with Li-Cheng Tsai, Manuel Cabezas, Andrey Sarantsev and Vladas Sidoravicius.

ThursdayJun 11, 201511:05
Geometric Functional Analysis and Probability SeminarRoom 261
Speaker:Anton MalyshevTitle:Metric distortion between random finite subsets of the intervalAbstract:opens in new windowin html    pdfopens in new window

Consider a random finite metric space X given by sampling n points in the unit interval uniformly, and a deterministic finite metric space U given by placing n points in the unit interval at uniform distance. With high probability, X will contain some pairs of points at distance roughly 1/n^2, so any bijection from X to U must distort distances by a factor of roughly n. However, with high probability, two of these random spaces, X_1 and X_2, have a bijection which distorts distances by a factor of only about n^2/3. The exponent of 2/3 is optimal.

ThursdayJun 04, 201511:05
Geometric Functional Analysis and Probability SeminarRoom 261
Speaker:Mark RudelsonTitle:Approximation complexity of convex bodiesAbstract:opens in new windowin html    pdfopens in new window
Consider the approximation of an n-dimensional convex body by a projection of a section of an N-dimensional simplex, and call the minimal N for which such approximation exists the approximation complexity of the body. The reason for such strange definition lies in computer science. A projection of a section of a simplex is the feasible set of a linear programming problem, and so it can be efficiently generated. We will discuss how large the approximation complexity of different classes of convex bodies can be.
ThursdayMay 28, 201511:05
Geometric Functional Analysis and Probability SeminarRoom 261
Speaker:Gerard Ben ArousTitle:The ant in the labyrinth: recent progressAbstract:opens in new windowin html    pdfopens in new window
ThursdayMay 21, 201511:05
Geometric Functional Analysis and Probability SeminarRoom 261
Speaker:Lenya RyzhikTitle:The weakly random Schroedinger equation: a consumer reportAbstract:opens in new windowin html    pdfopens in new window
Consider a Schroedinger equation with a weakly random time-independent potential. When the correlation function of the potential is, roughly speaking, of the Schwartz class, it has been shown by Spohn (1977), and Erdos and Yau (2001) that the kinetic limit holds -- the expectation of the phase space energy density of the solution converges weakly (after integration against a test function, not in the probabilistic sense) to the solution of a kinetic equation. We "extend" this result to potentials whose correlation functions satisfy (in some sense) "sharp" conditions, and also prove a parallel homogenization result for slowly varying initial conditions. I will explain the quotation marks above and make some speculations on the genuinely sharp conditions on the random potential that separate various regimes. This talk is a joint work with T. Chen and T. Komorowski
WednesdayMay 20, 201511:00
Geometric Functional Analysis and Probability SeminarRoom 108
Speaker:Kate JuschenkoTitle:Amenability of subgroups of interval exchange transformation groupAbstract:opens in new windowin html    pdfopens in new window
ThursdayMay 07, 201511:05
Geometric Functional Analysis and Probability SeminarRoom 261
Speaker:Ehud FriedgutTitle:An information theoretic proof of a hypercontractive inequalityAbstract:opens in new windowin html    pdfopens in new window

In the famous KKL (Kahn-Kalai-Linial) paper of 1988 the authors "imported" to combinatorics and theoretical computer science a hypercontractive inequality known as Beckner's ineqaulity (proven first, independently, by Gross and Bonami). This inequality has since become an extremely useful and influential tool, used in tens of papers, in a wide variety of settings. In many cases there are no proofs known that do not use the inequality.

In this talk I'll try to illuminate the information theoretic nature of both the inequality and its dual, touch upon a proof of the dual version from about a decade ago, (joint with V. Rodl), and a fresh (and unrelated) information theoretic proof of the primal version.

No prior knowledge will be assumed regarding discrete Fourier analysis, Entropy, and hypercontractivity.

ThursdayApr 16, 201511:05
Geometric Functional Analysis and Probability SeminarRoom 261
Speaker:Piotr MilosTitle:Extremal individuals in branching systemsAbstract:opens in new windowin html    pdfopens in new window

Branching processes have been subject of intense and fascinating studies for a long time. In my talk I will present two problems in order to highlight their rich structure and various technical approaches in the intersection of probability and analysis.
Firstly, I will present results concerning a branching random walk with the time-inhomogeneous branching law. We consider a system of particles, which at the end of each time unit produce offspring randomly and independently. The branching law, determining the number and locations of the offspring is the same for all particles in a given generation. Until recently, a common assumption was that the branching law does not change over time. In a pioneering work, Fang and Zeitouni (2010) considered a process with two macroscopic time intervals with different branching laws. In my talk I will present the results when the branching law varies at mesoscopic and microscopic scales. In arguably the most interesting case, when the branching law is sampled randomly for every step, I will present a quenched result with detailed asymptotics of the maximal particle. Interestingly, the disorder has a slowing-down effect manifesting itself on the log level.
Secondly, I will turn to the classical branching Brownian motion. Let us assume that particles move according to a Brownian motion with drift μ and split with intensity 1. It is well-know that for μ≥2√ the system escapes to infinity, thus the overall minimum is well-defined. In order to understand it better, we modify the process such that the particles are absorbed at position 0. I will present the results concerning the law of the number of absorbed particles N. In particular I will concentrate on P(N=0) and the maximal exponential moment of N. This reveals new deep connections with the FKPP equation. Finally, I will also consider −2√<μ<2√ and Nxt the number of particles absorbed until the time t when the system starts from x. In this case I will show the convergence to the traveling wave solution of the FKPP equation for an appropriate choice of x,t−>∞.
The results were obtained jointly with B. Mallein and with J. Berestycki, E. Brunet and S. Harris respectively.

ThursdayMar 26, 201511:05
Geometric Functional Analysis and Probability SeminarRoom 261
Speaker:Tom HutchcroftTitle:Hyperbolic and Parabolic Random MapsAbstract:opens in new windowin html    pdfopens in new window
We establish a sharp division of infinite random planar graphs into two types, hyperbolic and parabolic, showing that many probabilistic and geometric properties of such a graph are determined by the graph's average curvature, a local quantity which is often easy to compute. Work in progress with Omer Angel, Asaf Nachmias and Gourab Ray.
ThursdayMar 26, 201511:05
Geometric Functional Analysis and Probability SeminarRoom 261
Speaker:Tom HutchcroftTitle:Hyperbolic and Parabolic Random MapsAbstract:opens in new windowin html    pdfopens in new window
We establish a sharp division of infinite random planar graphs into two types, hyperbolic and parabolic, showing that many probabilistic and geometric properties of such a graph are determined by the graph's average curvature, a local quantity which is often easy to compute. Work in progress with Omer Angel, Asaf Nachmias and Gourab Ray.
ThursdayMar 12, 201511:05
Geometric Functional Analysis and Probability SeminarRoom 261
Speaker:Toby JohnsonTitle:The frog model on treesAbstract:opens in new windowin html    pdfopens in new window

Imagine that every vertex of a graph contains a sleeping frog. At time 0, the frog at some designated vertex wakes up and begins a simple random walk. When it lands on a vertex, the sleeping frog there wakes up and begins its own simple random walk, which in turn wakes up any sleeping frogs it lands on, and so on. This process is called the frog model.

I'll talk about a question posed by Serguei Popov in 2003: On an infinite d-ary tree, is the frog model recurrent or transient? That is, is each vertex visited infinitely or finitely often by frogs? The answer is that it depends on d: there's a phase transition between recurrence and transience as d grows. Furthermore, if the system starts with Poi(m) sleeping frogs on each vertex independently, there's a phase transition as m grows. This is joint work with Christopher Hoffman and Matthew Junge.

ThursdayFeb 12, 201511:05
Geometric Functional Analysis and Probability SeminarRoom 261
Speaker:Chaim Even ZoharTitle:Invariants of Random Knots and LinksAbstract:opens in new windowin html    pdfopens in new window
We study random knots and links in R^3 using the Petaluma model, which is based on the petal projections developed by Adams et al. (2012). In this model we obtain a formula for the distribution of the linking number of a random two-component link. We also obtain formulas for the expectations and the higher moments of the Casson invariant and the order-three knot invariant v3. These are the first precise formulas given for the distributions of invariants in any model for random knots or links. All terms above will be defined and explained. Joint work with Joel Hass, Nati Linial, and Tahl Nowik.
ThursdayFeb 05, 201511:05
Geometric Functional Analysis and Probability SeminarRoom 261
Speaker:Yuval PeledTitle:On the phase transition in random simplicial complexesAbstract:opens in new windowin html    pdfopens in new window
It is well-known that the model of random graphs undergoes a dramatic change around p=1/n. It is here that the random graph is, almost surely, no longer a forest, and here it first acquires a giant connected component. Several years ago, Linial and Meshulam have introduced the X_d(n,p) model, a probability space of n-vertex d-dimensional simplicial complexes, where X_1(n,p) coincides with G(n,p). Within this model we prove a natural d-dimensional analog of these graph theoretic phenomena. Specifically, we determine the exact threshold for the nonvanishing of the real d-th homology of complexes from X_d(n,p), and show that it is strictly greater than the threshold of d-collapsibility. In addition, we compute the real Betti numbers, i.e. the dimension of the homology groups, of X_d(n,p)for p=c/n. Finally, we establish the emergence of giant shadow at this threshold. (For d=1 a giant shadow and a giant component are equivalent). Unlike the case for graphs, for d > 1 the emergence of the giant shadow is a first order phase transition. The talk will contain the necessary toplogical backgorund on simplicial complexes, and will focus on the main idea of the proof: the local weak limit of random simplicial complexes and its role in the analysis of phase transitions. Joint work with Nati Linial.
ThursdayJan 29, 201511:05
Geometric Functional Analysis and Probability SeminarRoom 261
Speaker:Gady KozmaTitle:Random walk in random environment: the operator theory approachAbstract:opens in new windowin html    pdfopens in new window

Examine random walk in a stationary, ergodic, random environment which is bistochastic i.e. the sum of probabilities to enter any fixed vertex is 1. Consider the drift as a function on the probability space on the environments, and assume it belongs to domain of definition of where D is the symmetrized generator of the walk (this is the famous  The Actual Formula condition). We show that under these conditions the walk satisfies a central limit theorem. The proof uses the "relaxed sector condition" which shows an unexpected connection to the spectral theory of unbounded operators.

All terms will be explained in the talk. This is joint work with Balint Toth.

ThursdayJan 22, 201511:05
Geometric Functional Analysis and Probability SeminarRoom 261
Speaker:Thomas LebleTitle:Large deviations for the empirical field of Coulomb and Riesz systemsAbstract:opens in new windowin html    pdfopens in new window
We study a system of $N$ particles with Coulomb/Riesz pairwise interactions under a confining potential. After rescaling we deal with a microscopic quantity, the associated empirical point process, for which we give a large deviation principle whose rate function is the sum of a relative entropy and of the "renormalized energy" defined by Sandier-Serfaty. We also present applications to point processes emerging from random matrix theory. This is joint work with S. Serfaty.
ThursdayDec 18, 201410:30
Geometric Functional Analysis and Probability SeminarRoom 261
Speaker:Alexander FishTitle:Plunnecke inequalities in countable abelian groups - general caseAbstract:opens in new windowin html    pdfopens in new windowPLEASE NOTE UNUSUAL TIME
Plunnecke inequalities for sumsets of finite sets in abelian groups are extended to measure -preserving systems (mps). For a set A in a group, and a set B of positive measure in mps, we estimate the measure of the union of translations along the set A of B. To prove the new inequalities we extend the graph-theoretic method recently developed by Petridis to "measure graphs". As an application, through Furstenberg's correspondence principle, we obtain the new Plunnecke type inequalities for lower and upper Banach density in countable abelian groups. Based on joint works with M. Bjorklund, Chalmers, and with Kamil Bulinski, Sydne
ThursdayAug 28, 201411:00
Geometric Functional Analysis and Probability SeminarRoom 261
Speaker:Laurie FieldTitle:Two-sided radial SLE and length-biased chordal SLEAbstract:opens in new windowin html    pdfopens in new windowFilter test
Models in statistical physics often give measures on self-avoiding paths. We can restrict such a measure to the paths that pass through a marked point, obtaining a "pinned measure". The aggregate of the pinned measures over all possible marked points is just the original measure biased by the path's length. Does the analogous result hold for SLE curves, which appear in the scaling limits of many such models at criticality? We show that it does: the aggregate of two-sided radial SLE is length-biased chordal SLE, where the path's length is measured in the natural parametrisation.