Random Combinatorial Billiards and Stoned Exclusion Processes

TL;DR

This paper introduces random combinatorial billiard trajectories influenced by hyperplane reflections, analyzes associated Markov chains called stoned exclusion processes, and explores their stationary distributions and related growth processes.

Contribution

It develops a new framework connecting combinatorial billiards with Markov chains having known stationary distributions, and introduces the novel scan ASEP variant.

Findings

Stationary distributions linked to ASEP polynomials

Limit directions for billiard trajectories determined

New random growth processes for n-core partitions

Abstract

We introduce and study several random combinatorial billiard trajectories. Such a system, which depends on a fixed parameter $p\in(0,1)$, models a beam of light that travels in a Euclidean space, occasionally randomly reflecting off of a hyperplane in the Coxeter arrangement of an affine Weyl group with some probability that depends on the side of the hyperplane that it hits. In one case, we (essentially) recover Lam's reduced random walk in the limit as $p$ tends to $0$. The investigation of our random billiard trajectories relies on an analysis of new finite Markov chains that we call stoned exclusion processes. These processes have remarkable stationary distributions determined by well-studied polynomials such as ASEP polynomials, inhomogeneous TASEP polynomials, and open boundary ASEP polynomials; in many cases, it was previously not known how to construct Markov chains with these stationary distributions. Using multiline queues, we analyze correlations in the stoned multispecies TASEP, allowing us to determine limit directions for reduced random billiard trajectories and limit shapes for new random growth processes for $n$-core partitions. Our perspective coming from combinatorial billiards naturally leads us to formulate a new variant of the ASEP on $\mathbb{Z}$ called the scan ASEP, which we deem interesting in its own right.