Universal distribution of Lyapunov exponents for products of Ginibre matrices

TL;DR

This paper analytically derives the distribution of Lyapunov exponents for products of Ginibre matrices, showing they are normally distributed for large products and revealing a surprising equivalence between singular values and eigenvalue moduli.

Contribution

It provides a novel analytical derivation of Lyapunov exponent distributions for Ginibre matrix products, including their normality and variance, and uncovers the equivalence between singular values and eigenvalue radii in this context.

Findings

Lyapunov exponents are normally distributed for large product sizes.

Singular values and eigenvalue moduli share the same Lyapunov exponents asymptotically.

Derived a Gaussian approximation from asymptotic expansion of a Meijer G-function.

Abstract

Starting from exact analytical results on singular values and complex eigenvalues of products of independent Gaussian complex random $N\times N$ matrices also called Ginibre ensemble we rederive the Lyapunov exponents for an infinite product. We show that for a large number $t$ of product matrices the distribution of each Lyapunov exponent is normal and compute its $t$-dependent variance as well as corrections in a $1/t$ expansion. Originally Lyapunov exponents are defined for singular values of the product matrix that represents a linear time evolution. Surprisingly a similar construction for the moduli of the complex eigenvalues yields the very same exponents and normal distributions to leading order. We discuss a general mechanism for $2\times 2$ matrices why the singular values and the radii of complex eigenvalues collapse onto the same value in the large-$t$ limit. Thereby we rederive Newman's triangular law which has a simple interpretation as the radial density of complex eigenvalues in the circular law and study the commutativity of the two limits $t\to\infty$ and $N\to\infty$ on the global and the local scale. As a mathematical byproduct we show that a particular asymptotic expansion of a Meijer G-function with large index leads to a Gaussian.