Probability and Stochastic Analysis - University of Bonn

Abstract of "Random matrices and determinantal processes"

The aim of this work is to explain some connections between random matrices and determinantal processes. First we consider the eigenvalue distributions of the classical Gaussian random matrices ensembles. Of particular interest is the distribution of their largest eigenvalue in the limit of large matrices. For the Gaussian Unitary Ensemble, GUE, it is known as GUE Tracy-Widom distribution and appears in a lot of different models in combinatorics, growth models, equilibrium statistical mechanics, and in non-colliding random walks or Brownian particles. Secondly we introduce the determinantal processes, which are point processes which n-point correlation functions are given by a determinants of a kernel of an integral operator. It turns out that the eigenvalue distribution of the GUE random matrices is a determinantal process which kernel has a particular structure. This is the reason why the GUE Tracy-Widom distribution appears in a lot of models which are not related with random matrices.

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