Random Subwords and Billiard Walks in Affine Weyl Groups
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| Format: | Preprint |
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2025
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| _version_ | 1866916572266758144 |
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| author | Defant, Colin Jiradilok, Pakawut Mossel, Elchanan |
| author_facet | Defant, Colin Jiradilok, Pakawut Mossel, Elchanan |
| contents | 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 \[\lim_{K\to\infty}\frac{1}{\sqrt{K}}\mathbb{E}[\ell(v_p(\mathsf{b}^K))]=\sqrt{\frac{2}πr(r+1)\frac{p}{1-p}}.\] |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_11095 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Random Subwords and Billiard Walks in Affine Weyl Groups Defant, Colin Jiradilok, Pakawut Mossel, Elchanan Probability Combinatorics 05E16, 20F55, 60G50, 60J10 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 \[\lim_{K\to\infty}\frac{1}{\sqrt{K}}\mathbb{E}[\ell(v_p(\mathsf{b}^K))]=\sqrt{\frac{2}πr(r+1)\frac{p}{1-p}}.\] |
| title | Random Subwords and Billiard Walks in Affine Weyl Groups |
| topic | Probability Combinatorics 05E16, 20F55, 60G50, 60J10 |
| url | https://arxiv.org/abs/2501.11095 |