CLT for Circular beta-Ensembles at high temperature
CLT for Circular beta-Ensembles at high temperature
复制标题
高温下圆形 β 系综的 CLT
DOI：
10.1016/j.jfa.2020.108869
复制
发表时间：
2019
影响因子：
1.7
通讯作者：
Gaultier Lambert
中科院分区：
文献类型：
--
作者：
A. Hardy;Gaultier Lambert
我们考虑了高温下圆β-Englance的宏观大N极限，以及它的加权形式，在温度的倒数标度为β/N的区域中，对于某些参数β> 0。更准确地说，在极限N→∞，这个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=∞）之间插值的泛函的唯一极小值。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，其协方差结构介于Lebesgue L2（β= 0）和Sobolev H1/2（β=∞）范数之间.此外，我们得到的收敛速度的波动在W 2度量。我们的证明使用正常的近似结果Lambert等人。[24]，Chafaetan等人的库仑输运不等式。[8]以及对与所述限制协方差结构相关联的算子的谱分析。
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 β/N for some parameter β> 0. More precisely, in the limit N→∞, the equilibrium measure of this particle system is described as the unique minimizer of a functional which interpolates between the relative entropy (β= 0) and the weighted logarithmic energy (β=∞). 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 (β= 0) and the Sobolev H 1/2 (β=∞) 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 et al.[24], the Coulomb transport inequality of Chafaï et al.[8], and a spectral analysis for the operator associated with the limiting covariance structure.