Local eigenvalue statistics of one-dimensional random non-selfadjoint pseudo-differential operators

TL;DR

This paper studies the local eigenvalue statistics of one-dimensional nonselfadjoint semiclassical pseudo-differential operators with small random perturbations, revealing universal behavior described by Gaussian Analytic Functions.

Contribution

It demonstrates that the local spectral statistics are universal, depending only on spectral density and perturbation type, not on the specific distribution of the perturbations.

Findings

Spectral density follows a semiclassical Weyl law

Local eigenvalue statistics are universal

Statistics described by Gaussian Analytic Functions

Abstract

We consider a class of one-dimensional nonselfadjoint semiclassical pseudo-differential operators, subject to small random perturbations, and study the statistical properties of their (discrete) spectra, in the semiclassical limit $h\to 0$. We compare two types of random perturbation: a random potential vs. a random matrix. Hager and Sj\"ostrand had shown that, with high probability, the local spectral density of the perturbed operator follows a semiclassical form of Weyl's law, depending on the value distribution of the principal symbol of our pseudodifferential operator. Beyond the spectral density, we investigate the full local statistics of the perturbed spectrum, and show that it satisfies a form of universality: the statistical only depends on the local spectral density, and of the type of random perturbation, but it is independent of the precise law of the perturbation. This local statistics can be described in terms of the Gaussian Analytic Function, a classical ensemble of random entire functions.