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

TL;DR

This paper analyzes the maximum deviation of the Sine$_eta$ process, linking it to log-correlated Gaussian fields and the behavior of Gaussian $eta$-ensembles, providing insights into their extreme value statistics.

Contribution

It offers a direct analysis of the stochastic sine equation to describe the continuum limit of Pr"ufer phases in Gaussian $eta$-ensembles, connecting to log-correlated fields.

Findings

Maximum deviation aligns with log-correlated Gaussian field predictions

Analysis of the stochastic sine equation describes Pr"ufer phases

Results relate to the imaginary part of log-characteristic polynomials

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 Pr\"ufer phases of a Gaussian $\beta$-ensemble matrix.