A vector-valued almost sure invariance principle for Sinai billiards with random scatterers

TL;DR

This paper establishes a vector-valued almost sure invariance principle for a simplified model of Sinai billiards with randomly updated scatterers, accommodating weak dependence and non-stationarity in configurations.

Contribution

It introduces a novel invariance principle for time-dependent Sinai billiards, extending previous results to non-stationary and weakly dependent scatterer configurations.

Findings

Proves a vector-valued almost sure invariance principle for the model.

Provides an expression for the covariance matrix in non-stationary cases.

Improves accuracy and generality of existing invariance principles for Sinai billiards.

Abstract

Understanding the statistical properties of the aperiodic planar Lorentz gas stands as a grand challenge in the theory of dynamical systems. Here we study a greatly simplified but related model, proposed by Arvind Ayyer and popularized by Joel Lebowitz, in which a scatterer configuration on the torus is randomly updated between collisions. Taking advantage of recent progress in the theory of time-dependent billiards on the one hand and in probability theory on the other, we prove a vector-valued almost sure invariance principle for the model. Notably, the configuration sequence can be weakly dependent and non-stationary. We provide an expression for the covariance matrix, which in the non-stationary case differs from the traditional one. We also obtain a new invariance principle for Sinai billiards (the case of fixed scatterers) with time-dependent observables, and improve the accuracy and generality of existing results.