On the largest Lyapunov exponent for products of Gaussian matrices

TL;DR

This paper introduces a new integral formula for the largest Lyapunov exponent of Gaussian matrices across real, complex, and quaternion cases, enabling asymptotic analysis and comparisons in different models.

Contribution

It provides a novel integral formula for the largest Lyapunov exponent applicable to various types of Gaussian matrices, facilitating asymptotic and comparative studies.

Findings

Derived asymptotic expressions for large matrices

Compared Lyapunov exponents in spike and no-spike models

Unified formula applicable to real, complex, and quaternion cases

Abstract

The paper provides a new integral formula for the largest Lyapunov exponent of Gaussian matrices, which is valid in the real, complex and quaternion-valued cases. This formula is applied to derive asymptotic expressions for the largest Lyapunov exponent when the size of the matrix is large and compare the Lyapunov exponents in models with a spike and no spikes.