Moderate-to-large deviation asymptotics for real eigenvalues of the elliptic Ginibre matrices

TL;DR

This paper investigates the probabilities of rare events concerning the number of real eigenvalues in elliptic Ginibre matrices, bridging the gap between typical fluctuations and extreme deviations, and extends findings to classical ensembles.

Contribution

It derives moderate-to-large deviation probabilities for real eigenvalues in elliptic Ginibre matrices, connecting known fluctuation regimes with large deviations, including new results for classical ensembles.

Findings

Derived probabilities of rare events in moderate-to-large deviations regimes.

Connected Gaussian fluctuation regime with large deviation regime.

Extended results to classical real Ginibre ensemble.

Abstract

We study the statistics of the number of real eigenvalues in the elliptic deformation of the real Ginibre ensemble. As the matrix dimension grows, the law of large numbers and the central limit theorem for the number of real eigenvalues are well understood, but the probabilities of rare events remain largely unexplored. Large deviation type results have been obtained only in extreme cases, when either a vanishingly small proportion of eigenvalues are real or almost all eigenvalues are real. Here, in both the strong and weak asymmetry regimes, we derive the probabilities of rare events in the moderate-to-large deviation regime, thereby providing a natural connection between the previously known regime of Gaussian fluctuations and the large deviation regime. Our results are new even for the classical real Ginibre ensemble.