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 $N\times N$ form a determinantal point process. Near the soft edge, as both $M$ and $N$ go to infinity in such a way that $M/N\to \alpha$, $\alpha>0$, 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 $(a,+\infty)$. We derive a Tracy-Widom-like formula in terms of the unique solution of a certain matrix Riemann-Hilbert problem of size $2 \times 2$. The right-tail asymptotics for this solution is obtained by the Deift-Zhou non-linear steepest descent analysis.