Products of random matrices from fixed trace and induced Ginibre ensembles

TL;DR

This paper studies the eigenvalue distributions of fixed trace and mixed products of induced Ginibre matrices, revealing universality properties and providing explicit density correlation functions at finite sizes.

Contribution

It introduces a fixed trace constraint in the induced Ginibre ensemble, computes correlation functions, and analyzes the spectral density of products of such matrices, establishing universality classes.

Findings

Fixed trace ensemble density correlation functions computed at finite size.

Spectral density of mixed products derived via inverse Laplace transform.

Universality class of mixed product spectral density matches that of independent induced Ginibre matrices.

Abstract

We investigate the microcanonical version of the complex induced Ginibre ensemble, by introducing a fixed trace constraint for its second moment. Like for the canonical Ginibre ensemble, its complex eigenvalues can be interpreted as a two-dimensional Coulomb gas, which are now subject to a constraint and a modified, collective confining potential. Despite the lack of determinantal structure in this fixed trace ensemble, we compute all its density correlation functions at finite matrix size and compare to a fixed trace ensemble of normal matrices, representing a different Coulomb gas. Our main tool of investigation is the Laplace transform, that maps back the fixed trace to the induced Ginibre ensemble. Products of random matrices have been used to study the Lyapunov and stability exponents for chaotic dynamical systems, where the latter are based on the complex eigenvalues of the product matrix. Because little is known about the universality of the eigenvalue distribution of such product matrices, we then study the product of $m$ induced Ginibre matrices with a fixed trace constraint - which are clearly non-Gaussian - and $M-m$ such Ginibre matrices without constraint. Using an $m$-fold inverse Laplace transform, we obtain a concise result for the spectral density of such a mixed product matrix at finite matrix size, for arbitrary fixed $m$ and $M$. Very recently local and global universality was proven by the authors and their coworker for a more general, single elliptic fixed trace ensemble in the bulk of the spectrum. Here, we argue that the spectral density of mixed products is in the same universality class as the product of $M$ independent induced Ginibre ensembles.