Exponential bounds for the support convergence in the Single Ring Theorem

TL;DR

This paper establishes exponential bounds on the support convergence rate in the Single Ring Theorem for certain random matrices, improving error bounds and providing new convergence rate estimates.

Contribution

It provides exponential bounds for support convergence in the Single Ring Theorem, enhancing previous error estimates and analyzing the convergence rate.

Findings

Support convergence rate is at most n^{-1/6} log n.

Error bounds improved from polynomial to exponential decay.

Provides a simpler proof of support convergence under weaker assumptions.

Abstract

We consider an $n$ by $n$ matrix of the form $A=UTV$, with $U, V$ some independent Haar-distributed unitary matrices and $T$ a deterministic matrix. We prove that for $k\sim n^{1/6}$ and $b^2:=\frac{1}{n}\operatorname{Tr}(|T|^2)$, as $n$ tends to infinity, we have $$\mathbb{E} \operatorname{Tr} (A^{k}(A^{k})^*) \ \lesssim \ b^{2k}\qquad \textrm{and} \qquad\mathbb{E}[|\operatorname{Tr} (A^{k})|^2] \ \lesssim \ b^{2k}.$$ This gives a simple proof (with slightly weakened hypothesis) of the convergence of the support in the Single Ring Theorem, improves the available error bound for this convergence from $n^{-\alpha}$ to $e^{-cn^{1/6}}$ and proves that the rate of this convergence is at most $n^{-1/6}\log n$.