Extremal laws for the real Ginibre ensemble

TL;DR

This paper proves that the scaled spectral radius of large real Ginibre matrices converges to a Gumbel distribution, extending known results from complex and quaternion cases, and introduces a new form for the largest real eigenvalue distribution.

Contribution

It establishes the Gumbel law for the spectral radius of the real Ginibre ensemble and presents a novel characterization of the largest real eigenvalue's limit law.

Findings

Spectral radius converges to Gumbel distribution as matrix size grows

New form for the limit law of the largest real eigenvalue

Results align with known behaviors in complex and quaternion ensembles

Abstract

The real Ginibre ensemble refers to the family of $n\times n$ matrices in which each entry is an independent Gaussian random variable of mean zero and variance one. Our main result is that the appropriately scaled spectral radius converges in law to a Gumbel distribution as $n\rightarrow\infty$. This fact has been known to hold in the complex and quaternion analogues of the ensemble for some time, with simpler proofs. Along the way we establish a new form for the limit law of the largest real eigenvalue.