2016/05/02 by N. S. Witte, Peter J. Forrester, Witte, N. S. +1 · 1 citation
Mathematics · Physics and Astronomy · #15B52 #33E17 #34E05 #34M56 #60K35 #62E15 #Classical Analysis and ODEs (math.CA) #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Mathematical functions and polynomials #Nonlinear Waves and Solitons #Random Matrices and Applications
paper · pdf · doi:10.48550/arxiv.1605.00704
openalex publication_date 2016/05/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 FM, 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 ≥ 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.