Lyapunov exponents for truncated unitary and Ginibre matrices

TL;DR

This paper investigates the asymptotic behavior of Lyapunov exponents for products of truncated Haar unitary and Ginibre matrices, revealing a 'picket-fence' spacing pattern linked to multiplicative Brownian motion.

Contribution

It introduces a novel connection between random matrix products and multiplicative Brownian motion, providing new insights into the spectral statistics of these matrices.

Findings

Lyapunov exponents follow 'picket-fence' spacing asymptotically

Connection established between matrix products and multiplicative Brownian motion

Contour integral formulas used to derive spectral statistics

Abstract

In this note, we show that the Lyapunov exponents of mixed products of random truncated Haar unitary and complex Ginibre matrices are asymptotically given by equally spaced `picket-fence' statistics. We discuss how these statistics should originate from the connection between random matrix products and multiplicative Brownian motion on $\operatorname{GL}_n(\mathbb{C})$, analogous to the connection between discrete random walks and ordinary Brownian motion. Our methods are based on contour integral formulas for products of classical matrix ensembles from integrable probability.