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 ${}_0{F}_M$, 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 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 generalised this formalism to general $M\ge 1$, 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 4-th 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 4-th order equation are discussed, as is a possible relationship of the $M=2$ systems to so-called 4-dimensional Painlevé-type equations.

arXiv comment33 pages, 4 figures, 1 table

References in corpus (9)

• Biorthogonal ensembles
• Application of the $\tau$-Function Theory of Painlevé Equations to Random Matrices: PIV, PII and the GUE
• Products of Rectangular Random Matrices: Singular Values and Progressive Scattering
• On the Numerical Evaluation of Distributions in Random Matrix Theory: A Review
• Eigenvalue statistics for product complex Wishart matrices
• The spectral curve of the Eynard-Orantin recursion via the Laplace transform
• Differential equations for singular values of products of Ginibre random matrices
• Degeneration scheme of 4-dimensional Painlevé-type equations
• Muttalib--Borodin ensembles in random matrix theory --- realisations and correlation functions

Cited by in corpus (2)

• On Wright's generalized Bessel kernel
• Large gap asymptotics at the hard edge for product random matrices and Muttalib-Borodin ensembles