On the maximum of the CβE field

Abstract

In this article, we investigate the extremal values of (the logarithm of) the characteristic polynomial of a random unitary matrix whose spectrum is distributed according to the circular beta ensemble (C$\beta$E). More precisely, assuming that $X_n$ is this characteristic polynomial and $\mathbb{U}$ is the unit circle, we prove that

$${sup\hspace{0.17em}}_{z\in \mathbb{U}}\Re log{X}_n\left(z\right)=\sqrt{\frac{2}{\beta }}(logn-\frac{3}{4}loglogn+\mathcal{O}(1\left)\right),$$ as well as an analogous statement for the imaginary part. The notation $\mathcal{O}\left(1\right)$ 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 $\beta =2$ case, which corresponds to the circular unitary ensemble (CUE) field.