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We consider the propagation of microwave photons along an array of superconducting grains with a set of weakly-coupled grains at its center. Quantum fluctuations of charge on the weakly-coupled grains make the process of “photon splitting” effective. In such a process, an incoming photon may be split into a number of photons of lower energy. The minimal number of photons so created depends on the symmetry properties of the corresponding quantum impurity model. As an example, we consider a specific circuit allowing quantum fluctuations between two charge configurations of two weakly-coupled grains, thus mimicking the behavior of an anisotropic Kondo impurity. We relate the total rate of conversion of incoming photons into the lower-energy ones to the linear dynamic spin susceptibility of the Kondo model. The spectral distribution of the outgoing photons yields information about higher-order local correlations in the quantum impurity dynamics. Finally, we reveal an interesting relation between this problem and transport along the edge of a 2D topological insulator with a magnetic impurity.

Models that are able to reproduce the abundance and clustering of galaxies are based on a statistical match between halos (or subhalos) and galaxies at a given redshift. The observational data are then used to constrain the mass relation between halos and galaxies. These models have been widely used to predict cosmological parameters, star formation rates, merger rates, weak lensing signal, and to interpret the results of hydrodynamical simulations. I will first introduce and discuss this approach, pointing out its advantages and limitations. I will then argue that the evolution of satellite galaxies is the main unknown in these models, and is probably more complicated than what is usually assumed. For example, the stellar mass of a satellite galaxy might depend not only on its host subhalo mass, but also on the mass of its group, and on the time it first became a satellite. A new methodology will be presented that is capable of exploring the complex behavior of satellite galaxies in a relatively simple manner. Using this new approach we were able to compute the correlation functions for a very large number of models (~10^7), finding a large range of accepted models, much larger than previously claimed. The new method is useful for various clustering studies, including AGNs, HI gas, and high-redshift galaxies.