标题： 关于C$β$E场的最大值 显示英文标题

标题： On the maximum of the C$β$E field

摘要： 在本文中，我们研究了随机酉矩阵的特征多项式（对数）的极值，其谱分布遵循圆型Beta系综（C$\beta$E）。更准确地说，如果$X_n$是这个特征多项式，$\mathbb{U}$是单位圆，我们证明了：$$\sup_{z \in \mathbb{U} } \Re \log X_n(z) = \sqrt{\frac{2}{\beta}} \left(\log n - \frac{3}{4} \log \log n + \mathcal{O}(1) \right)\ ,$$以及关于虚部的类似结论。符号$\mathcal{O}(1)$表示由$n$索引的相应随机变量族是紧致的。 这回答了Fyodorov、Hiary和Keating的一个猜想，该猜想最初是针对$\beta$等于$2$的情况提出的，这对应于CUE场。 显示英文摘要

摘要： 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.