We consider the macroscopic large N limit of the Circular beta-Ensemble at high temperature, and its weighted version as well, in the regime where the inverse temperature scales as beta/N for some parameter beta>0. More precisely, in the large N limit, the equilibrium measure of this particle system is described as the unique minimizer of a functional which interpolates between the relative entropy (beta=0) and the weighted logarithmic energy (beta=\infty). More precisely, we provide subGaussian concentration estimates in the W1 metric for the deviations of the empirical measure to this equilibrium mesure. The purpose of this work is to show that the fluctuation of the empirical measure around the equilibrium measure converges towards a Gaussian field whose covariance structure interpolates between the Lebesgue L^2 (beta=0) and the Sobolev H^{1/2} (beta=\infty) norms. We furthermore obtain a rate of convergence for the fluctuations in the W_2 metric. Our proof uses the normal approximation result of Lambert, Ledoux and Webb [2017] the Coulomb transport inequality of Chafai, Hardy, Maida [2018] and a spectral analysis for the operator associated with the limiting covariance structure.

中文翻译：

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高温下圆形β-集合的CLT

我们考虑了在高温下圆形 beta-Ensemble 的宏观大 N 极限及其加权版本，在逆温标为 beta/N 的情况下，某些参数 beta>0。更准确地说，在大 N 限制中，该粒子系统的平衡度量被描述为在相对熵 (beta=0) 和加权对数能量 (beta=\infty) 之间插值的函数的唯一最小值。更准确地说，我们在 W1 度量中提供了亚高斯浓度估计，用于经验度量与该平衡度量的偏差。这项工作的目的是表明，围绕均衡测度的经验测度的波动收敛于一个高斯场，其协方差结构在 Lebesgue L^2 (beta=0) 和 Sobolev H^{1/2} 之间进行插值（ beta=\infty) 规范。此外，我们还获得了 W_2 度量波动的收敛速度。我们的证明使用了 Lambert、Ledoux 和 Webb [2017] 的正态近似结果、Chafai、Hardy、Maida [2018] 的库仑输运不等式以及对与极限协方差结构相关的算子的谱分析。