# Integrable structure of products of finite complex Ginibre random matrices

## TL;DR

This paper explores the integrable structure of the squared singular values of products of finite complex Ginibre matrices, deriving Hamiltonian systems and differential equations that describe their gap probabilities.

## Contribution

It generalizes the Hamiltonian structure and integrable systems approach to finite matrix products, extending previous results from the infinite case to finite matrices.

## Key findings

- Finite kernel expressed in integrable form.
- Hamiltonian structure for finite matrices established.
- Coupled differential equations for gap probabilities derived.

## Abstract

We consider the squared singular values of the product of $M$ standard complex Gaussian matrices. Since the squared singular values form a determinantal point process with a particular Meijer G-function kernel, the gap probabilities are given by a Fredholm determinant based on this kernel. It was shown by Strahov \cite{St14} that a hard edge scaling limit of the gap probabilities is described by Hamiltonian differential equations which can be formulated as an isomonodromic deformation system similar to the theory of the Kyoto school. We generalize this result to the case of finite matrices by first finding a representation of the finite kernel in integrable form. As a result we obtain the Hamiltonian structure for a finite size matrices and formulate it in terms of a $(M+1) \times (M+1)$ matrix Schlesinger system. The case $M=1$ reproduces the Tracy and Widom theory which results in the Painlev\'e V equation for the $(0,s)$ gap probability. Some integrals of motion for $M = 2$ are identified, and a coupled system of differential equations in two unknowns is presented which uniquely determines the corresponding $(0,s)$ gap probability.