Local Single Ring Theorem

TL;DR

This paper extends the Single Ring Theorem to a local scale, providing microscopic eigenvalue distribution results for large matrices with prescribed singular values, and introduces new subordination results for non-Hermitian matrices.

Contribution

It presents a local version of the Single Ring Theorem at microscopic scales and develops a matrix subordination result for sums of non-Hermitian matrices.

Findings

Proves local eigenvalue distribution at scale $( ext{log } N)^{-1/4}$.

Establishes a local law for singular values of sums of non-Hermitian matrices.

Provides delocalization results for singular vectors.

Abstract

The Single Ring Theorem, by Guionnet, Krishnapur and Zeitouni, describes the empirical eigenvalue distribution of a large generic matrix with prescribed singular values, i.e. an $N\times N$ matrix of the form $A=UTV$, with $U, V$ some independent Haar-distributed unitary matrices and $T$ a deterministic matrix whose singular values are the ones prescribed. In this text, we give a local version of this result, proving that it remains true at the microscopic scale $(\log N)^{-1/4}$. On our way to prove it, we prove a matrix subordination result for singular values of sums of non Hermitian matrices, as Kargin did for Hermitian matrices. This allows to prove a local law for the singular values of the sum of two non Hermitian matrices and a delocalization result for singular vectors.