Random Subwords and Billiard Walks in Affine Weyl Groups

TL;DR

The article analyzes random subwords in irreducible affine Weyl groups via a geometric random-billiard model, proving a central limit theorem for the position of the associated alcove after many steps. It derives a computable, representation-theory–driven formula for the limiting covariance parameter $\sigma_\mathsf{b}^2$, including a simple closed form when the word $\mathsf{b}$ contains a single occurrence of the distinguished reflection $s_0$, and provides explicit values for all affine types when $\mathsf{b}$ is a Coxeter word. A corollary yields asymptotics for the expected Coxeter length $\mathbb{E}[\ell(v_p(\mathsf{b}^K))]$, with a concrete limit in the $\widetilde A_r$ case with $\mathsf{b}$ using every simple reflection. The work connects probabilistic limit theorems with the combinatorics of affine Weyl groups, offering a computational framework that unifies and extends prior results on random subwords and Demazure products, and highlighting links to random walks and potentially to TASEP-like processes. These results provide a new lens on the asymptotic geometry of alcoves under stochastic dynamics parameterized by $p$ and the starting word $\mathsf{b}$.

Abstract

Let $W$ be an irreducible affine Weyl group, and let $\mathsf{b}$ be a finite word over the alphabet of simple reflections of $W$. Fix a probability $p\in(0,1)$. For each integer $K\geq 0$, let $\mathsf{sub}_p(\mathsf{b}^K)$ be the random subword of $\mathsf{b}^K$ obtained by deleting each letter independently with probability $1-p$. Let $v_p(\mathsf{b}^K)$ be the element of $W$ represented by $\mathsf{sub}_p(\mathsf{b}^K)$. One can view $v_p(\mathsf{b}^K)$ geometrically as a random alcove; in many cases, this alcove can be seen as the location after a certain amount of time of a random billiard trajectory that, upon hitting a hyperplane in the Coxeter arrangement of $W$, reflects off of the hyperplane with probability $1-p$. We show that the asymptotic distribution of $v_p(\mathsf{b}^K)$ is a central spherical multivariate normal distribution with some variance $σ_{\mathsf{b}}^2$ depending on $\mathsf{b}$ and $p$. We provide a formula to compute $σ_{\mathsf{b}}^2$ that is remarkably simple when $\mathsf{b}$ contains only one occurrence of the simple reflection that is not in the associated finite Weyl group. As a corollary, we provide an asymptotic formula for $\mathbb{E}[\ell(v_p(\mathsf{b}^K))]$, the expected Coxeter length of $v_p(\mathsf{b}^K)$. For example, when $W=\widetilde A_{r}$ and $\mathsf{b}$ contains each simple reflection exactly once, we find that \[\lim_{K\to\infty}\frac{1}{\sqrt{K}}\mathbb{E}[\ell(v_p(\mathsf{b}^K))]=\sqrt{\frac{2}πr(r+1)\frac{p}{1-p}}.\]

Figures (3)

• Figure 1: The trajectory of a random billiard walk corresponding to a random subword $\mathsf{sub}_{4/5}((s_2s_1s_0)^{200})$ in type $\widetilde{A}_2$. Line segments traversed more frequently are drawn darker. The red dot indicates the starting point $\mathbf{z}_0$.
• Figure 2: The trajectory of a random billiard walk corresponding to a random subword $\mathsf{sub}_{4/5}((s_1s_2s_1s_0)^{150})$ in type $\widetilde{C}_2$. Line segments traversed more frequently are drawn darker. The red dot indicates the starting point $\mathbf{z}_0$.
• Figure 3: The trajectory of the billiard walk corresponding to the subword $s_2s_1s_0s_2\,\underline{\,\,\,\,\,}\,\underline{\,\,\,\,\,}\,s_2s_1\,\underline{\,\,\,\,\,}\,s_2\,\underline{\,\,\,\,\,}\,s_0s_2s_1s_0$ of $(s_2s_1s_0)^5$ in type $\widetilde{A}_2$. The red dot indicates the starting point $\mathbf{z}_0$.

Theorems & Definitions (36)

• Example 1.1
• Remark 1.2
• Remark 1.3
• Theorem 1.4
• Corollary 1.5
• Corollary 1.6
• Theorem 1.7
• Theorem 1.8
• Remark 1.9
• Theorem 1.10