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

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: $$\underset{z\in \mathbb{U}}{\sup }\Re \log X_n(z)=\sqrt{\frac{2}{\beta }}\left(\log n-\frac{3}{4}\log \log n+\mathcal{O}(1)\right),$$ 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.

74 pages ; v1: Preliminary version; v2: Submitted version

References in corpus (4)

• Maximum of the characteristic polynomial of random unitary matrices
• Matrix models for circular ensembles
• Derrida's random energy models. From spin glasses to the extremes of correlated random fields
• The subleading order of two dimensional cover times

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