Random Subwords and Billiard Walks in Affine Weyl Groups

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 $K\ge 0$, let ${\mathsf{s}\mathsf{u}\mathsf{b}}_p({\mathsf{b}}^K)$ 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 ${\mathsf{s}\mathsf{u}\mathsf{b}}_p({\mathsf{b}}^K)$. 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 ${\sigma }_{\mathsf{b}}^2$ depending on $\mathsf{b}$ and $p$. We provide a formula to compute ${\sigma }_{\mathsf{b}}^2$ 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 $\mathbb{E}[\mathrm{\ell }(v_p({\mathsf{b}}^K))]$, the expected Coxeter length of $v_p({\mathsf{b}}^K)$. For example, when $W={\tilde{A}}_r$ and $\mathsf{b}$ contains each simple reflection exactly once, we find that $$\underset{K\to \mathrm{\infty }}{\lim }\frac{1}{\sqrt{K}}\mathbb{E}[\mathrm{\ell }(v_p({\mathsf{b}}^K))]=\sqrt{\frac{2}{\pi }r(r+1)\frac{p}{1-p}}\mathrm{.}$$