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On local point process limits of random matrices

The study of random matrices goes back to the works of Wishart (1920’s) and Wigner (1930’s).
At the time they introduced a special class of models that had explicitly computable joint density
functions for their eigenvalues. These joint densities turn out to be specific cases of a more
general physical model called Coulomb gas models which describe particles interacting through
some Hamiltonian. In this talk we will discuss a specific class of these Coulomb gas models
called beta-ensembles which are a partial generalization of the random matrix eigenvalue
process. I will begin by introducing a random matrix model and the generalization of its
eigenvalue process to a beta-ensemble. We will then discuss how one can study the interactions
of individual eigenvalues as the number of them grows to infinity. I will introduce the Sine-beta
process, one of the limit processes that appears when the eigenvalues are rescaled to see these
“local” interactions, and discuss several results and techniques that may be used for studying this
process.