The log-Characteristic Polynomial of Generalized Wigner Matrices is Log-Correlated

TL;DR

This paper demonstrates that the logarithm of the characteristic polynomial of generalized Wigner matrices converges to a log-correlated field in large dimensions, revealing universal fluctuation behavior at the spectrum's edge.

Contribution

It extends the log-correlation results to a wider class of models and the spectrum edge, using a combination of local laws, stochastic analysis, and CLT techniques.

Findings

Logarithm of characteristic polynomial converges to a log-correlated field

Universal fluctuation behavior at spectrum edge

Established local laws and Wegner estimates at microscopic scales

Abstract

We prove that in the limit of large dimension, the distribution of the logarithm of the characteristic polynomial of a generalized Wigner matrix converges to a log-correlated field. In particular, this shows that the limiting joint fluctuations of the eigenvalues are also log-correlated. Our argument mirrors that of \cite{BouMod2019}, which is in turn based on the three-step argument of \cite{ErdPecRmSchYau2010,ErdSchYau2011Uni}, but applies to a wider class of models, and at the edge of the spectrum. We rely on (i) the results in the Gaussian cases, special cases of the results in \cite{BouModPai2021}, (ii) the local laws of \cite{ErdYauYin2012}(iii) the observable \cite{Bou2020} introduced and its analysis of the stochastic advection equation this observable satisfies, and (iv) the argument for a central limit theorem on mesoscopic scales in \cite{LanLopSos2021}. For the proof, we also establish a Wegner estimate and local law down to the microscopic scale, both at the edge of the spectrum.