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 KM(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 KM(x, y) χJ(y), where J is a disjoint union of intervals, J=uj(a2j-1, a2j), and χJ, 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 KM(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.
|Original language||American English|
|Journal||Journal of Physics A: Mathematical and Theoretical|
|State||Published - 2014|
Bibliographical notePublisher Copyright:
© 2014 IOP Publishing Ltd.
- determinantal point processes
- integrable differential equations
- products of random matrices