arXiv 1909.011422019-09math.PRmath-phmath.FAmath.MP

高温下圆形β系综的CLT

CLT for Circular beta-Ensembles at High Temperature

Adrien Hardy, Gaultier Lambert

备注：: A section added about the continuity of the variance as the inverse temperature beta tends to 0 or infinity. References updated

摘要：

我们考虑在高温下的宏观大N极限的圆形β-Englance，以及它的加权版本，在该制度的逆温度标度为β/N的某些参数β>0。更确切地说，在大N极限下，该粒子系统的平衡测度被描述为在相对熵（β =0）和加权对数能量（β = \infty）之间插值的泛函的唯一极小化。更确切地说，我们提供了subGaussian浓度估计在W1度量的偏差的经验措施，这个平衡measure。本文的目的是证明经验测度在平衡测度附近的涨落收敛于一个高斯场，该高斯场的协方差结构介于勒贝格L^2（beta=0）和索伯列夫H^{1/2}（beta= \infty）范数之间。我们还得到了W_2度量涨落的一个收敛速率。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与极限协方差结构相关的算子的谱分析。

英文摘要：

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.

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