On minimal singular values of random matrices with correlated entries

被引：11

作者：

Goetze, F. ^{[1
]}

Naumov, A. ^{[1
,2
]}

Tikhomirov, A. ^{[3
]}

机构：

[1] Univ Bielefeld, Fac Math, Bielefeld, Germany

[2] Moscow MV Lomonosov State Univ, Fac Computat Math & Cybernet, Moscow, Russia

[3] RAS, Dept Math, Komi Res Ctr, Ural Branch, Syktyvkar, Russia

来源：

RANDOM MATRICES-THEORY AND APPLICATIONS | 2015年 / 4卷 / 02期

关键词：

Random matrices; circular law; elliptic law; non-identically distributed entries; Stieltjes transform; UNIVERSALITY;

D O I：

10.1142/S2010326315500069

中图分类号：

O4 [物理学];

学科分类号：

0702 ;

摘要：

Let X be a random matrix whose pairs of entries X-jk and X-kj are correlated and vectors (X-jk, X-kj), for 1 <= j < k <= n, are mutually independent. Assume that the diagonal entries are independent from off-diagonal entries as well. We assume that EXjk = 0, EX (2)(jk) = 1, for any j, k = 1,..., n and EXjkXkj = rho for 1 <= j < k = n. Let M-n be a non-random nxn matrix with parallel to M-n parallel to <= K-n(Q), for some positive constants K > 0 and Q >= 0. Let s(n) (X + M-n) denote the least singular value of the matrix X + M-n. It is shown that there exist positive constants A and B depending on K, Q, rho only such that P(sn(X + M-n) <= n (A)) = n (B). As an application of this result we prove the elliptic law for this class of matrices with non-identically distributed correlated entries.

引用

页数：30

共 21 条

[1]

Akemann G., 2011, The Oxford Handbook of Random Matrix Theory

[2]

Bentkus V., 2003, LIET MAT RINK, V43, P444

[3] Around the circular law [J].