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Title: Random matrices and the Gaussian multiplicative chaos on the line

Abstract: The Gaussian multiplicative chaos is a relatively new universal object in probability that has many interesting geometric properties.

The characteristic polynomial of many classes of random matrices is, in many cases conjecturally, one class of finite approximation to these random measures. Great progress has been made on showing the random matrices from specific ``circular ensembles’’ converge to the GMC. Likewise, some progress has been made for unitarily—invariant random matrices. We show some new partial progress in showing the "Gaussian beta—ensemble" has a GMC limit. This we do by using the representation of its characteristic polynomial as an entry in a product of independent random two--by--two matrices. For a point z in the complex plane, at which the transfer matrix is to be evaluated, this product of transfer matrices splits into three independent factors, each of which can be understood as a different dynamical system in the complex plane. Using this, we show that the characteristic polynomial is always represented as product of at most three terms, two of which are exponentials of Gaussian fields and one of which is the stochastic Airy function, up to vanishing multiplicative errors. Sufficiently far into the complex plane, it can be represented using only one. At a point z at the spectral edge, there are two. At a point z in the bulk of the spectrum, all three are necessary to describe the characteristic polynomial. Joint work with Gaultier Lambert.