Differential equations for singular values of products of Ginibre random matrices

It was proved by Akemann et al (2013 Phys. Rev. E 88 052118) 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 Meijers G-functions. Kuijlaars and Zhang (arXiv:1308.1003) 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) χ_J(y), where J is a disjoint union of intervals, J=u_j(a_2j-1, a_2j), and χ_J, and 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 the Hamiltonian differential associated with the corresponding Fredholm determinant, and relate them with the monodromy preserving deformation equations of the Jimbo, Miwa, Môri, 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 nonlinear ordinary differential equations.