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

Diane Holcomb; Elliot Paquette

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arXiv · 1801.08989

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

Diane Holcomb·Elliot Paquette

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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.

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Diane Holcomb, Elliot Paquette. 2018-01-26. The maximum deviation of the Sine$_\beta$ counting process. https://arxiv.org/abs/1801.08989

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