Fluctuations of the Stieltjes transform of the empirical spectral distribution of selfadjoint polynomials in Wigner and deterministic diagonal matrices

TL;DR

This paper studies the fluctuations of the Stieltjes transform of spectral distributions of selfadjoint polynomials in Wigner and diagonal matrices, showing convergence to a Gaussian process with covariance linked to operator-valued functions.

Contribution

It introduces a new analysis of spectral fluctuations for selfadjoint polynomials in random and deterministic matrices, connecting to operator-valued subordination functions.

Findings

Fluctuations converge to a centered complex Gaussian process.

Covariance is expressed via operator-valued subordination functions.

Results apply to a broad class of selfadjoint polynomials.

Abstract

We investigate the fluctuations around the mean of the Stieltjes transform of the empirical spectral distribution of any selfadjoint noncommutative polynomial in a Wigner matrix and a deterministic diagonal matrix. We obtain the convergence in distribution to a centred complex Gaussian process whose covariance is expressed in terms of operator-valued subordination functions.