Local single ring theorem on optimal scale

TL;DR

This paper proves that the eigenvalue distribution of certain random matrices converges locally on the optimal scale within the bulk of the spectrum, extending the single ring theorem to finer scales with optimal rates.

Contribution

It establishes the optimal local eigenvalue distribution convergence for the single ring theorem in the bulk regime, improving understanding of spectral behavior at microscopic scales.

Findings

Convergence of empirical eigenvalue distribution on scale N^{-1/2+ε}

Optimal convergence rate established

Results hold for both unitary and orthogonal Haar measures

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 non-negative deterministic $N$ by $N$ matrix. The single ring theorem [26] 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, i.e. 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 $O(N)$.