Gap probability for products of random matrices in the critical regime

TL;DR

This paper investigates the gap probability in the critical regime for the singular values of products of Ginibre matrices, deriving a Tracy-Widom-like formula via Riemann-Hilbert analysis.

Contribution

It introduces a new Tracy-Widom-like formula for the gap probability in the critical regime of product matrices, using a matrix Riemann-Hilbert problem approach.

Findings

Derived a Tracy-Widom-like formula for gap probability.

Obtained right-tail asymptotics using Deift-Zhou steepest descent.

Characterized the limiting determinantal process 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.