Differential equations for singular values of products of Ginibre random matrices

It was proved by Akemann, Ipsen and Kieburg that squared singular values of products of $M$ complex Ginibre random matrices form a determinantal point process whose correlation kernel is expressible in terms of Meijer's $G$-functions. Kuijlaars and Zhang recently showed that at the edge of the spectrum, this correlation kernel has a remarkable scaling limit $K_M(x,y)$ which can be understood as a generalization of the classical Bessel kernel of Random Matrix Theory. In this paper we investigate the Fredholm determinant of the operator with the kernel $K_M(x,y)\chi_J(y)$, where $J$ is a disjoint union of intervals, $J=\cup_j(a_{2j-1},a_{2j})$, and $\chi_J$ is the characteristic function of the set $J$. This Fredholm determinant is equal to the probability that $J$ contains no particles of the limiting determinantal point process defined by $K_M(x,y)$ (the gap probability). We derive Hamiltonian differential equations associated with the corresponding Fredholm determinant, and relate them with the monodromy preserving deformation equations of the Jimbo, Miwa, Mori, Ueno and Sato theory. In the special case $J=(0,s)$ we give a formula for the gap probability in terms of a solution of a system of non-linear ordinary differential equations.