Lyapunov exponents for products of complex Gaussian random matrices

TL;DR

This paper derives exact formulas for the Lyapunov exponents of products of complex Gaussian random matrices, including sums of exponents and cases involving diffusing matrices, advancing understanding in random matrix theory.

Contribution

It provides explicit formulas for Lyapunov exponents of complex Gaussian matrix products, a novel exact calculation in this domain.

Findings

Exact Lyapunov exponents for complex Gaussian matrix products

Explicit expression for sum of Lyapunov exponents in complex and real cases

Lyapunov exponents for diffusing complex matrices

Abstract

The exact value of the Lyapunov exponents for the random matrix product $P_N = A_N A_{N-1}...A_1$ with each $A_i = \Sigma^{1/2} G_i^{\rm c}$, where $\Sigma$ is a fixed $d \times d$ positive definite matrix and $G_i^{\rm c}$ a $d \times d$ complex Gaussian matrix with entries standard complex normals, are calculated. Also obtained is an exact expression for the sum of the Lyapunov exponents in both the complex and real cases, and the Lyapunov exponents for diffusing complex matrices.