Zooming in on local level statistics by supersymmetric extension of free probability

TL;DR

This paper introduces a supersymmetric extension of free probability to analyze local spectral statistics in random matrix ensembles, providing a new method for studying universality in level correlations.

Contribution

It extends free probability techniques using supersymmetry to investigate local spectral statistics, especially level correlations, in random matrix models.

Findings

Established universality in a stochastic scattering matrix model.

Developed a supersymmetric approach linking free probability and local statistics.

Provided a new analytical tool for level correlation analysis.

Abstract

We consider unitary ensembles of Hermitian NxN matrices H with a confining potential NV where V is analytic and uniformly convex. From work by Zinn-Justin, Collins, and Guionnet and Maida it is known that the large-N limit of the characteristic function for a finite-rank Fourier variable K is determined by the Voiculescu R-transform, a key object in free probability theory. Going beyond these results, we argue that the same holds true when the finite-rank operator K has the form that is required by the Wegner-Efetov supersymmetry method of integration over commuting and anti-commuting variables. This insight leads to a potent new technique for the study of local statistics, e.g., level correlations. We illustrate the new technique by demonstrating universality in a random matrix model of stochastic scattering.