Products of Independent Quaternion Ginibre Matrices and their Correlation Functions

TL;DR

This paper analyzes the eigenvalue correlations of products of independent quaternion Ginibre matrices, revealing that their joint eigenvalue distribution matches that of a single Ginibre matrix with a complex weight, and computes correlation functions for finite and large matrices.

Contribution

It introduces the eigenvalue correlation functions for products of quaternion Ginibre matrices and derives skew-orthogonal polynomials for their complex weight functions.

Findings

Eigenvalue distribution matches that of a single Ginibre matrix with a modified weight.

Explicit correlation functions are computed for finite matrix sizes.

Large matrix limit reveals macroscopic and microscopic density behaviors.

Abstract

We discuss the product of independent induced quaternion ($\beta=4$) Ginibre matrices, and the eigenvalue correlations of this product matrix. The joint probability density function for the eigenvalues of the product matrix is shown to be identical to that of a single Ginibre matrix, but with a more complicated weight function. We find the skew-orthogonal polynomials corresponding to the weight function of the product matrix, and use the method of skew-orthogonal polynomials to compute the eigenvalue correlation functions for product matrices of finite size. The radial behavior of the density of states is studied in the limit of large matrices, and the macroscopic density is discussed. The microscopic limit at the origin, at the edge(s) and in the bulk is also discussed for the radial behavior of the density of states.