The Product of m Real N x N Ginibre Matrices: Real Eigenvalues in the Critical Regime m = O(N)

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 -> infinity 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 -> infinity 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 -> infinity and alpha -> 0, respectively.