A central limit theorem with explicit Lyapunov exponent and variance for products of $2\times2$ random non-invertible matrices

TL;DR

This paper derives explicit formulas for the Lyapunov exponent and a central limit theorem with variance for products of i.i.d. non-invertible 2x2 matrices, with applications to physical models like the random Hill's equation.

Contribution

It provides the first explicit formulas for Lyapunov exponents and CLT variance for non-invertible matrix products under mild conditions.

Findings

Explicit formulas for Lyapunov exponent and CLT variance derived.

Examples with exact Lyapunov exponent and variance computed.

Applications to physical models like the random Hill's equation.

Abstract

The theory of products of random matrices and Lyapunov exponents have been widely studied and applied in the fields of biology, dynamical systems, economics, engineering and statistical physics. We consider the product of an i.i.d. sequence of $2\times 2$ random non-invertible matrices with real entries. Given some mild moment assumptions we prove an explicit formula for the Lyapunov exponent and prove a central limit theorem with an explicit formula for the variance in terms of the entries of the matrices. We also give examples where exact values for the Lyapunov exponent and variance are computed. An important example where non-invertible matrices are essential is the random Hill's equation, which has numerous physical applications, including the astrophysical orbit problem.