Gap probability for products of random matrices in the critical regime

The singular values of a product of M independent Ginibre matrices of size N x N form a determinantal point process. Near the soft edge, as both M and N go to infinity in such a way that M/N -> alpha, alpha > 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, +infinity). We derive a Tracy-Widom-like formula in terms of the unique solution of a certain matrix Riemann- Hilbert problem of size 2 x 2. The right-tail asymptotics for this solution is obtained by the Deift-Zhou non-linear steepest descent analysis. (c) 2021 Elsevier Inc. All rights reserved.