Taming Irrationality: An Invariance Principle for the Random Billiard Walk

TL;DR

This paper extends the understanding of the random billiard walk by proving a central limit theorem for all initial directions and demonstrating convergence to isotropic Brownian motion, revealing new probabilistic behaviors.

Contribution

It introduces a comprehensive invariance principle for the random billiard walk, generalizing previous results to all initial directions and establishing convergence to Brownian motion.

Findings

Convergence of the rescaled process to isotropic Brownian motion.

Extension of the CLT to all initial directions.

Continuous variation of the limiting covariance in parameters.

Abstract

The random billiard walk is a stochastic process $(L_t)_{t\geq 0}$ in which a laser moves through the Coxeter arrangement of an affine Weyl group in $\mathbb{R}^d$, reflecting at each hyperplane with probability $p\in (0, 1)$ and transmitting unchanged otherwise. Defant, Jiradilok, and Mossel introduced this process from the perspective of algebraic combinatorics and established that, for initial directions aligned with the coroot lattice, $L_t/\sqrt{t}$ converges to a centered spherical Gaussian. We bring analytic tools from ergodic theory and probability to the problem and extend this central limit theorem to all initial directions. More strongly, we prove the rescaled trajectories $t\mapsto n^{-1/2}L_{tn}$ converge to isotropic Brownian motion. Away from directions with rational dependencies, the limiting covariance varies continuously in $p$ and the initial direction.