Upper tail large deviations for extremal eigenvalues of the real, complex and symplectic elliptic Ginibre matrices

TL;DR

This paper analyzes the probabilities of extreme eigenvalues in large elliptic Ginibre matrices across different symmetry classes, providing unified asymptotic formulas for large deviations of spectral radius and rightmost eigenvalues.

Contribution

It introduces a unified framework for asymptotic large deviation probabilities of extremal eigenvalues in elliptic Ginibre ensembles, covering real, complex, and symplectic cases.

Findings

Asymptotic formulas for large deviation probabilities of spectral radius.

Asymptotic behavior of the rightmost eigenvalue.

Unified approach applicable to different symmetry classes.

Abstract

We consider the elliptic Ginibre ensembles in the real, complex and symplectic symmetry classes. As the matrix size tends to infinity, we derive the asymptotic behaviour of the upper tail large deviation probabilities for both the spectral radius and the rightmost eigenvalue. More generally, we obtain asymptotic formulas for the probability that an eigenvalue is found in a prescribed region outside the support of the elliptic law, thereby providing a unified framework in which the results for the spectral radius and the rightmost eigenvalue appear as special cases. The key ingredient of our analysis is the precise asymptotic behaviour of the associated one-point functions, which is of independent interest.