Fluctuations and correlations for products of real asymmetric random matrices

TL;DR

This paper investigates the statistical behavior of real eigenvalues in products of real Ginibre matrices, establishing their asymptotic normality, universality, and detailed fluctuation properties across different spectral regimes.

Contribution

It provides the first comprehensive analysis of real eigenvalue statistics for products of real Ginibre matrices, including central limit theorems, correlation functions, and universality results.

Findings

Asymptotic normality of linear eigenvalue statistics

Explicit limiting variance in global and mesoscopic regimes

Universal correlation functions in bulk, edge, and origin regimes

Abstract

We study the real eigenvalue statistics of products of independent real Ginibre random matrices. These are matrices all of whose entries are real i.i.d. standard Gaussian random variables. For such product ensembles, we demonstrate the asymptotic normality of suitably normalised linear statistics of the real eigenvalues and compute the limiting variance explicitly in both global and mesoscopic regimes. A key part of our proof establishes uniform decorrelation estimates for the related Pfaffian point process, thereby allowing us to exploit weak dependence of the real eigenvalues to give simple and quick proofs of the central limit theorems under quite general conditions. We also establish the universality of these point processes. We compute the asymptotic limit of all correlation functions of the real eigenvalues in the bulk, origin and spectral edge regimes. By a suitable strengthening of the convergence at the edge, we also obtain the limiting fluctuations of the largest real eigenvalue. Near the origin we find new limiting distributions characterising the smallest positive real eigenvalue.