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

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• The Fyodorov-Hiary-Keating Conjecture. I
• On the moments of the moments of the characteristic polynomials of random unitary matrices
• Maximum and coupling of the sine-Gordon field
• Critical Gaussian multiplicative chaos: a review
• Statistics of extremes in eigenvalue-counting staircases
• Extreme values of CUE characteristic polynomials: a numerical study
• On the critical points of random matrix characteristic polynomials and of the Riemann $\xi$-function
• Asymptotics for averages over classical orthogonal ensembles
• Maxima of log-correlated fields: some recent developments
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• Band-limited mimicry of point processes by point processes supported on a lattice
• Dimension Results for the Spectral Measure of the Circular Beta Ensembles
• Strong approximation of Gaussian beta-ensemble characteristic polynomials: the hyperbolic regime
• Universality of the minimum modulus for random trigonometric polynomials
• Global rigidity and exponential moments for soft and hard edge point processes
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• Disordered statistical physics in low dimensions: extremes, glass transition, and localization
• On the moments of moments of random matrices and Ehrhart polynomials
• Multiscale coupling and the maximum of $\mathcal{P}(\varphi {)}_2$ models on the torus