RANDOM MATRICES: UNIVERSALITY OF LOCAL SPECTRAL STATISTICS OF NON-HERMITIAN MATRICES

• 成果类型: Article
• 署名作者: Tao, Terence; Vu, Van
• 署名单位: University of California System; University of California Los Angeles; Yale University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/13-AOP876
• 发表日期: 2015
• 页码: 782-874
• 关键词: LITTLEWOOD-OFFORD PROBLEM; bulk universality; condition number; gaussian fluctuations; eigenvalue statistics; tail probabilities; quadratic-forms; semicircle law; limit-theorem; distributions

摘要

It is a classical result of Ginibre that the normalized bulk k-point correlation functions of a complex n x n Gaussian matrix with independent entries of mean zero and unit variance are asymptotically given by the determinantal point process on C with kernel K-infinity (z, w):=1/pi e-vertical bar z vertical bar(2)/2-vertical bar w vertical bar(2)/2+z (w) over bar in the limit n ->infinity. In this paper, we show that this asymptotic law is universal among all random n x n matrices M-n whose entries are jointly independent, exponentially decaying, have independent real and imaginary parts and whose moments match that of the complex Gaussian ensemble to fourth order. Analogous results at the edge of the spectrum are also obtained. As an application, we extend a central limit theorem for the number of eigenvalues of complex Gaussian matrices in a small disk to these more general ensembles. These results are non-Hermitian analogues of some recent universality results for Hermitian Wigner matrices. However, a key new difficulty arises in the non-Hermitian case, due to the instability of the spectrum for such matrices. To resolve this issue, we the need to work with the log-determinants log vertical bar det(M-n - z(0)) vertical bar rather than with the Stieltjes transform 1/n tr(M-n - z(0))(-1) in order to exploit Girko's Hermitization method. Our main tools are a four moment theorem for these log-determinants, together with a strong concentration result for the log-determinants in the Gaussian case. The latter is established by studying the solutions of a certain nonlinear stochastic difference equation. With some extra consideration, we can extend our arguments to the real case, proving universality for correlation functions of real matrices which match the real Gaussian ensemble to the fourth order. As an application, we show that a real n x n matrix whose entries are jointly independent, exponentially decaying and whose moments match the real Gaussian ensemble to fourth order has root 2n/pi+o(root n) real eigenvalues asymptotically almost surely.