Dynamical correlation functions for products of random matrices

TL;DR

This paper introduces a family of discrete-time random processes based on products of random matrices, analyzes their correlation functions, and derives explicit formulas and scaling limits for special cases, advancing understanding of matrix product dynamics.

Contribution

It provides explicit formulas for dynamical correlation functions of matrix product processes and explores their scaling limits, extending previous static analyses to dynamic settings.

Findings

The process is a discrete-time determinantal point process.

Explicit correlation functions are derived for special matrix cases.

Hard edge scaling limits of kernels are identified.

Abstract

We introduce and study a family of random processes with a discrete time related to products of random matrices. Such processes are formed by singular values of random matrix products, and the number of factors in a random matrix product plays a role of a discrete time. We consider in detail the case when the (squared) singular values of the initial random matrix form a polynomial ensemble, and the initial random matrix is multiplied by standard complex Gaussian matrices. In this case we show that the random process is a discrete-time determinantal point process. For three special cases (the case when the initial random matrix is a standard complex Gaussian matrix, the case when it is a truncated unitary matrix, or the case when it is a standard complex Gaussian matrix with a source) we compute the dynamical correlations functions explicitly, and find the hard edge scaling limits of the correlation kernels. The proofs rely on the Eynard-Mehta theorem, and on contour integral representations for the correlation kernels suitable for an asymptotic analysis.