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×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→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,+∞). We derive a Tracy–Widom-like formula in terms of the unique solution of a certain matrix Riemann–Hilbert problem of size 2 × 2. The right-tail asymptotics for this solution is obtained by the Deift–Zhou non-linear steepest descent analysis.