Large \(n\) limit for the product of two coupled random matrices. (English) Zbl 1508.81829
Summary: For a pair of coupled rectangular random matrices we consider the squared singular values of their product, which form a determinantal point process. We show that the limiting mean distribution of these squared singular values is described by the second component of the solution to a vector equilibrium problem. This vector equilibrium problem is defined for three measures with an upper constraint on the first measure and an external field on the second measure. We carry out the steepest descent analysis for a \(4\times 4\) matrix-valued Riemann-Hilbert problem, which characterizes the correlation kernel and is related to mixed type multiple orthogonal polynomials associated with the modified Bessel functions. A careful study of the vector equilibrium problem, combined with this asymptotic analysis, ultimately leads to the aforementioned convergence result for the limiting mean distribution, an explicit form of the associated spectral curve, as well as local Sine, Meijer-G and Airy universality results for the squared singular values considered.
MSC:
| 81Q05 | Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics |
| 60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |
| 81Q50 | Quantum chaos |
| 82C44 | Dynamics of disordered systems (random Ising systems, etc.) in time-dependent statistical mechanics |
| 15B52 | Random matrices (algebraic aspects) |
| 35Q15 | Riemann-Hilbert problems in context of PDEs |
| 33C10 | Bessel and Airy functions, cylinder functions, \({}_0F_1\) |
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