Large deviation theorem for zeros of polynomials and Hermitian random matrices

TL;DR

This paper develops abstract large deviation theorems for polynomial zeros and applies them to Hermitian random matrices, providing new estimates for spectral distributions under moment constraints.

Contribution

It introduces abstract large deviation results for polynomial zeros and applies them to Hermitian matrices to improve spectral distribution estimates.

Findings

New large deviation estimates for zeros of polynomials.

Enhanced bounds for spectral distributions of Hermitian matrices.

Results depend on controlling the fourth moments of matrix entries.

Abstract

We give abstract versions of the large deviation theorem for the distribution of zeros of polynomials and apply them to the characteristic polynomials of Hermitian random matrices. We obtain new estimates related to the local semi-circular law for the empirical spectral distribution of these matrices when the 4th moments of their entries are controlled.