Probability of all eigenvalues real for products of standard Gaussian matrices

Abstract

With {Xi} independent N × N standard Gaussian random matrices, the probability that all eigenvalues are real for the matrix product Pm = XmXm − 1⋅⋅⋅X1 is expressed in terms of an N/2 × N/2 (N even) and (N + 1)/2 × (N + 1)/2 (N odd) determinant. The entries of the determinant are certain Meijer G-functions. In the case m = 2 high precision computation indicates that the entries are rational multiples of π2, with the denominator a power of 2, and that to leading order in N decays as . We are able to show that for general m and large N, with an explicit bm. An analytic demonstration that as m → ∞ is given.