## The Annals of Probability

### Local single ring theorem on optimal scale

#### 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 $O(N)$.

#### Article information

Source
Ann. Probab., Volume 47, Number 3 (2019), 1270-1334.

Dates
Revised: December 2017
First available in Project Euclid: 2 May 2019

https://projecteuclid.org/euclid.aop/1556784020

Digital Object Identifier
doi:10.1214/18-AOP1284

Mathematical Reviews number (MathSciNet)
MR3945747

Zentralblatt MATH identifier
07067270

#### Citation

Bao, Zhigang; Erdős, László; Schnelli, Kevin. Local single ring theorem on optimal scale. Ann. Probab. 47 (2019), no. 3, 1270--1334. doi:10.1214/18-AOP1284. https://projecteuclid.org/euclid.aop/1556784020

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#### Supplemental materials

• Supplement to “Local single ring theorem on optimal scale”. We establish the proof of Theorem 2.2 and the proof of Theorem 4.3 for large $\eta$. Moreover, in the appendices at the end we collect some auxiliary information.