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May 2019 Local single ring theorem on optimal scale
Zhigang Bao, László Erdős, Kevin Schnelli
Ann. Probab. 47(3): 1270-1334 (May 2019). DOI: 10.1214/18-AOP1284

Abstract

Let $U$ and $V$ be two independent $N$ by $N$ random matrices that are distributed according to Haar measure on $U(N)$. Let $\Sigma$ be a nonnegative deterministic $N$ by $N$ matrix. The single ring theorem [Ann. of Math. (2) 174 (2011) 1189–1217] asserts that the empirical eigenvalue distribution of the matrix $X:=U\Sigma V^{*}$ converges weakly, in the limit of large $N$, to a deterministic measure which is supported on a single ring centered at the origin in $\mathbb{C}$. Within the bulk regime, that is, in the interior of the single ring, we establish the convergence of the empirical eigenvalue distribution on the optimal local scale of order $N^{-1/2+\varepsilon}$ and establish the optimal convergence rate. The same results hold true when $U$ and $V$ are Haar distributed on $O(N)$.

Citation

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Zhigang Bao. László Erdős. Kevin Schnelli. "Local single ring theorem on optimal scale." Ann. Probab. 47 (3) 1270 - 1334, May 2019. https://doi.org/10.1214/18-AOP1284

Information

Received: 1 January 2017; Revised: 1 December 2017; Published: May 2019
First available in Project Euclid: 2 May 2019

zbMATH: 07067270
MathSciNet: MR3945747
Digital Object Identifier: 10.1214/18-AOP1284

Subjects:
Primary: 46L54, 60B20

Rights: Copyright © 2019 Institute of Mathematical Statistics

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Vol.47 • No. 3 • May 2019
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