Random Lie group actions on compact manifolds: A perturbative analysis

TL;DR

This paper analyzes the behavior of Birkhoff sums for random Lie group actions on compact manifolds, showing they approximate integrals with respect to a unique smooth measure under weak coupling conditions.

Contribution

It introduces a perturbative approach that replaces traditional irreducibility assumptions with effective coupling conditions for analyzing random Lie group actions.

Findings

Birkhoff sums approximate integrals with respect to a unique smooth measure.

Results apply to random matrix products and disordered quantum wire models.

Errors are proportional to the coupling constant.

Abstract

A random Lie group action on a compact manifold generates a discrete time Markov process. The main object of this paper is the evaluation of associated Birkhoff sums in a regime of weak, but sufficiently effective coupling of the randomness. This effectiveness is expressed in terms of random Lie algebra elements and replaces the transience or Furstenberg's irreducibility hypothesis in related problems. The Birkhoff sum of any given smooth function then turns out to be equal to its integral w.r.t. a unique smooth measure on the manifold up to errors of the order of the coupling constant. Applications to the theory of products of random matrices and a model of a disordered quantum wire are presented.