The maximum deviation of the $\text{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.