On the maximum of the C$\beta$E field

TL;DR

This paper determines the asymptotic maximum of the logarithm of the characteristic polynomial for matrices in the Circular Beta Ensemble, confirming a conjecture for all beta values.

Contribution

It extends the understanding of extremal values of characteristic polynomials from the CUE case to general Beta ensembles, providing precise asymptotics.

Findings

Asymptotic maximum of the log characteristic polynomial is proportional to 0 d7 (\,log n - rac{3}{4} \, log log n)

Results hold uniformly over the unit circle for all b2 > 0

Confirms a conjecture for the maximum of the Cb5E characteristic polynomial

Abstract

In this paper, we investigate the extremal values of (the logarithm of) the characteristic polynomial of a random unitary matrix whose spectrum is distributed according the Circular Beta Ensemble (C$\beta$E). More precisely, if $X_n$ is this characteristic polynomial and $\mathbb{U}$ the unit circle, we prove that: $$\sup_{z \in \mathbb{U} } \Re \log X_n(z) = \sqrt{\frac{2}{\beta}} \left(\log n - \frac{3}{4} \log \log n + \mathcal{O}(1) \right)\ ,$$ as well as an analogous statement for the imaginary part. The notation $\mathcal{O}(1)$ means that the corresponding family of random variables, indexed by $n$, is tight. This answers a conjecture of Fyodorov, Hiary and Keating, originally formulated for the case where $\beta$ equals to $2$, which corresponds to the CUE field.