The maximum deviation of the Sine${}_{\beta }$ counting process

Abstract

In this paper, we consider the maximum of the ${\text{Sine}}_{\beta }$ counting process from its expectation. We show the leading order behavior is consistent with the predictions of log-correlated Gaussian fields, also consistent with work on the imaginary part of the log-characteristic polynomial of random matrices. We do this by a direct analysis of the stochastic sine equation, which gives a description of the continuum limit of the Prüfer phases of a Gaussian $\beta$-ensemble matrix.

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