Large deviations for the extremal eigenvalues of Ginibre ensembles

TL;DR

This paper develops large deviation principles for the extremal eigenvalues of Ginibre ensembles, revealing that deviations in the real Ginibre case originate from eigenvalues on the real line and providing universal estimates for i.i.d. matrices.

Contribution

It establishes large deviation principles with explicit rate functions for extremal eigenvalues of Ginibre ensembles, including real cases and universal estimates for i.i.d. matrices.

Findings

Large deviation principles with explicit rate functions for Ginibre eigenvalues.

Deviation estimates for the second leading eigenvalue term.

Universality of polynomially small deviation estimates for i.i.d. matrices.

Abstract

We establish large deviation principles for the extremal eigenvalues of the Ginibre ensembles with good rate functions. In contrast to the typical estimates for the extremal eigenvalues, the large deviations for the real Ginibre ensemble come from the eigenvalues lying on the real line. Moreover, we also derive deviation estimates for the second leading term in the asymptotic expansion of the extremal eigenvalues. These polynomially small deviation estimates are universal for any i.i.d. matrices under a mild moment condition.