Singular Values of Products of Ginibre Random Matrices

Abstract

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 , also referred to as hyper‐Bessel functions. In the case 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 Painlevé 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 , but has not exhibited its reduction. After detailing the necessary working in the case , we consider the problem of reducing the 12 coupled differential equations in the case 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 systems to so‐called four‐dimensional Painlevé‐type equations.