Gap probability for products of random matrices in the critical regime

Abstract

The singular values of a product of M independent Ginibre matrices of size N ×N form a determinantal point process. As both M and N go to infinity in such a way that M/N → α, α > 0, a scaling limit emerges. We consider a gap probability for the corresponding limiting determinantal process, namely, the probability that there are no particles in the interval (a, +∞), a > 0. This probability is evaluated explicitly in terms of the unique solution of a certain matrix Riemann-Hilbert problem of size 2 × 2. The right-tail asymptotics of this solution is obtained by the Deift-Zhou non-linear steepest descent analysis.