Universality of random matrices and local relaxation flow
Universality of random matrices and local relaxation flow
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DOI：
10.1007/s00222-010-0302-7
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发表时间：
2011-07-01
影响因子：
3.1
通讯作者：
Yau, Horng-Tzer
中科院分区：
文献类型：
--
作者：
Erdos, Laszlo;Schlein, Benjamin;Yau, Horng-Tzer
考虑带参数β的Dyson布朗运动，其中β = 1，2，4对应于对称、厄米特和四元数自对偶系综特征值的特征值流.对于任意的β a/千分之一，我们证明了Dyson Brown运动到局部平衡的弛豫时间在N（-zeta）上有界，其中zeta > 0.证明是基于对戴森布朗运动w.r.t.的熵流的估计。一种“伪平衡测量”。作为这一估计的应用，我们证明了在极限N -> a下，N × N对称Wigner系综的谱块特征值间距统计与高斯正交系综（GOE）的谱块特征值间距统计相同.对维格纳系综的矩阵元素的概率分布的假设是一个次指数衰减和一些小的限制支持。
Consider the Dyson Brownian motion with parameter beta, where beta=1,2,4 corresponds to the eigenvalue flows for the eigenvalues of symmetric, hermitian and quaternion self-dual ensembles. For any beta a parts per thousand yen1, we prove that the relaxation time to local equilibrium for the Dyson Brownian motion is bounded above by N (-zeta) for some zeta > 0. The proof is based on an estimate of the entropy flow of the Dyson Brownian motion w.r.t. a "pseudo equilibrium measure". As an application of this estimate, we prove that the eigenvalue spacing statistics in the bulk of the spectrum for NxN symmetric Wigner ensemble is the same as that of the Gaussian Orthogonal Ensemble (GOE) in the limit N -> a. The assumptions on the probability distribution of the matrix elements of the Wigner ensemble are a subexponential decay and some minor restriction on the support.