Singular Values of Products of Ginibre Random Matrices

The squared singular values of the product of M complex Ginibre matrices form a biorthogonal ensemble, and thus their distribution is fully determined by a correlation kernel. The kernel permits a hard edge scaling to a form specified in terms of certain Meijer G-functions, or equivalently hypergeometric functions F-0(M), also referred to as hyper-Bessel functions. In the case M = 1, it is well known that the corresponding gap probability for no squared singular values in (0, s) can be evaluated in terms of a solution of a particular sigma form of the Painleve III' system. One approach to this result is a formalism due to Tracy and Widom, involving the reduction of a certain integrable system. Strahov has generalized this formalism to general M = 1, but has not exhibited its reduction. After detailing the necessary working in the case M = 1, we consider the problem of reducing the 12 coupled differential equations in the case M = 2 to a single differential equation for the resolvent. An explicit fourth-order nonlinear is found for general hard edge parameters. For a particular choice of parameters, evidence is given that this simplifies to a much simpler third-order nonlinear equation. The small and large s asymptotics of the fourth-order equation are discussed, as is a possible relationship of the M = 2 systems to so-called four-dimensional Painleve-type equations.