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Nick Simm

Title: Random matrix theory and log-correlated Gaussian fields

Abstract: I will describe recent results on relations between random matrix theory and log-correlated Gaussian fields. A rather famous example of the latter is the Gaussian Free Field, which appears in numerous areas of mathematical physics and probability. We will show that certain objects from random matrix theory converge to what look like one-dimensional "cuts" of this field, depending on the way the field is regularized. The main example will concern the log characteristic polynomial of a Hermitian random matrix, for which I will present Theorems describing the convergence in law on both mesoscopic and macroscopic scales. Given time, in the case of the GUE, I will present a (non-heuristic and conjectural) discussion on the extreme value statistics of the characteristic polynomial.