Spectrum and pseudospectrum for quadratic polynomials in Ginibre matrices

TL;DR

This paper proves that the spectral distribution of quadratic polynomials in large Ginibre matrices converges to a non-commutative Brown measure, using pseudospectrum control and anti-concentration properties of matrix-valued random walks.

Contribution

It establishes the convergence of empirical spectral distributions for quadratic polynomials in Ginibre matrices to their Brown measure, extending non-commutative spectral analysis.

Findings

Convergence of spectral distribution to Brown measure

Quantitative control on pseudospectrum of polynomial matrices

Identification of structural reasons affecting anti-concentration

Abstract

For a fixed quadratic polynomial $\mathfrak{p}$ in $n$ non-commuting variables, and $n$ independent $N\times N$ complex Ginibre matrices $X_1^N,\dots, X_n^N$, we establish the convergence of the empirical spectral distribution of $P^N =\mathfrak{p}(X_1^N,\dots, X_n^N)$ to the Brown measure of $\mathfrak{p}$ evaluated at $n$ freely independent circular elements $c_1,\dots, c_n$ in a non-commutative probability space. The main step of the proof is to obtain quantitative control on the pseudospectrum of $P^N$. Via the well-known linearization trick this hinges on anti-concentration properties for certain matrix-valued random walks, which we find can fail for structural reasons of a different nature from the arithmetic obstructions that were illuminated in works on the Littlewood--Offord problem for discrete scalar random walks.