The Product of $m$ real $N\times N$ Ginibre matrices: Real eigenvalues in the critical regime $m=O(N)$

TL;DR

This paper investigates the behavior of real eigenvalues in the product of $m$ real Ginibre matrices of size $N$, especially in the critical regime where $m$ scales linearly with $N$, revealing how eigenvalue statistics interpolate between known limits.

Contribution

It introduces a critical scaling regime for the product of real Ginibre matrices where $m$ is proportional to $N$, deriving eigenvalue statistics that connect previous asymptotic results.

Findings

Derived the expected number and variance of real eigenvalues in the critical regime

Established the rescaled density of real eigenvalues in the critical scaling

Interpolated between known eigenvalue statistics for fixed $m$ and large $m$ limits

Abstract

We study the product $P_m$ of $m$ real Ginibre matrices with Gaussian elements of size $N$, which has received renewed interest recently. Its eigenvalues, which are either real or come in complex conjugate pairs, become all real with probability one when $m\to\infty$ at fixed $N$. In this regime the statistics becomes deterministic and the Lyapunov spectrum has been derived long ago. On the other hand, when $N\to\infty$ and $m$ is fixed, it can be expected that away from the origin the same local statistics as for a single real Ginibre ensemble at $m=1$ prevails. Inspired by analogous findings for products of complex Ginibre matrices, we introduce a critical scaling regime when the two parameters are proportional, $m=\alpha N$. We derive the expected number, variance and rescaled density of real eigenvalues in this critical regime. This allows us to interpolate between previous recent results in the above mentioned limits when $\alpha\to\infty$ and $\alpha\to0$, respectively.