LOCAL INHOMOGENEOUS CIRCULAR LAW

被引：35

作者：

Alt, Johannes ^{[1
]}

Erdos, Laszlo ^{[1
]}

Krueger, Torben ^{[1
]}

机构：

[1] IST Austria, Campus 1, A-3400 Klosterneuburg, Austria

来源：

ANNALS OF APPLIED PROBABILITY | 2018年 / 28卷 / 01期

基金：

欧洲研究理事会;

关键词：

Circular law; local law; variance profile; RANDOM MATRICES; SPECTRAL STATISTICS; UNIVERSALITY;

D O I：

10.1214/17-AAP1302

中图分类号：

O21 [概率论与数理统计]; C8 [统计学];

学科分类号：

020208 ; 070103 ; 0714 ;

摘要：

We consider large random matrices X with centered, independent entries, which have comparable but not necessarily identical variances. Girko's circular law asserts that the spectrum is supported in a disk and in case of identical variances, the limiting density is uniform. In this special case, the local circular law by Bourgade et al. [Probab. Theory Related Fields 159 (2014) 545-595; Probab. Theory Related Fields 159 (2014) 619-660] shows that the empirical density converges even locally on scales slightly above the typical eigenvalue spacing. In the general case, the limiting density is typically inhomogeneous and it is obtained via solving a system of deterministic equations. Our main result is the local inhomogeneous circular law in the bulk spectrum on the optimal scale for a general variance profile of the entries of X.

引用

页码：148 / 203

页数：56

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