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

Abstract

We study the product Pm 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 → ∞ 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 → ∞ 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 = αN . 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 α → ∞ and α → 0, respectively.

Real Eigenvalues of Elliptic Random Matrices

“…(See [1] for analogous results for products of GinOE matrices.) For the GinOE (τ = 0), the central limit theorem for the number of real eigenvalues (or its linear statistics in general) was proved in [19,45].…”