Directional Extremal Statistics for Ginibre Eigenvalues

TL;DR

This paper provides a concise proof that the maximum real part of eigenvalues of large Ginibre matrices follows a Gumbel distribution and form a Poisson process, with implications for locating spectral maxima in large random matrices.

Contribution

It offers a direct proof with effective error bounds for the extremal eigenvalue distribution in Ginibre matrices, extending previous results and aiding spectral analysis of large matrices.

Findings

Maximum real parts follow Gumbel distribution

Eigenvalues form a Poisson point process asymptotically

Effective error bounds for correlation kernel

Abstract

We consider the eigenvalues of a large dimensional real or complex Ginibre matrix in the region of the complex plane where their real parts reach their maximum value. This maximum follows the Gumbel distribution and that these extreme eigenvalues form a Poisson point process, asymptotically as the dimension tends to infinity. In the complex case these facts have already been established by Bender \cite{MR2594353} and in the real case by Akemann and Phillips \cite{MR3192169} even for the more general elliptic ensemble with a sophisticated saddle point analysis. The purpose of this note is to give a very short direct proof in the Ginibre case with an effective error term. Moreover, our estimates on the correlation kernel in this regime serve as a key input for accurately locating $\max\Re\mathrm{Spec}(X)$ for any large matrix $X$ with i.i.d. entries in the companion paper \cite{2206.04448}.