Hitting Probabilities for Finite-Range Random Walks on Infinite Trees 

Speaker: Chevalier Guillaume (中国科学院数学与系统科学研究院)

Title: Hitting Probabilities for Finite-Range Random Walks on Infinite Trees 

Inviter: 随机分析研究中心

Language: English  

Time & Venue: 2026年4月3日（周五）16:00--17:00 南楼613

Abstract: When we consider a random walk on an infinite tree, it is natural to ask how the sequence of probabilities that such a random walk is in a given subset of vertices after a certain among of steps, behaves when the number of steps goes to infinity. In this talk, I will present a method for obtaining asymptotic expansions to any order for such sequence of probabilities, improving a result due to S. P. Lalley(Ann. Probab. 21 (1993), no. 4, p.2087-2130). More precisely, if $x,y$ are two vertices of an infinite, locally finite tree in which every vertex has at least three neighbours, and if $(Z_n)_n$ denotes an irreducible finite range random walk on the tree, based at $x$, whose transition kernel is invariant under a cofinite action of an automorphism group of the tree, then there exist constants $r\in\mathbb{Z}/d\mathbb{Z}$, $C>0$ and $(c_l)_{l\geq 1}$ such that we have the asymptotic expansion:

$$\mathbb{P}^x(Z_{dn+r}\in \{y\})\sim_\infty \frac{C}{R^{dn} n^{3/2}}\left( 1 + \sum_{l\geq 1} \frac{c_{l}}{n^{l}}\right),$$

and $\mathbb{P}^x(Z_{dn+t}=y)=0$ for any $t\neq r \, [d]$, with $d$ the period of the random walk and $R>1$ the radius of convergence of the associated generating function called Green's function. We will also see that such analogous asymptotic holds \textit{mutatis mutandis}, almost surely when we replace $\{y\}$ by a geodesic ray, or a horocycle for nearest-neighbour random walks.