The simple random walk on $\mathbb{Z}^p$ shows two drastically different behaviours depending on the value of $p$: it is recurrent when $p\in\{1,2\}$ while it escapes (with a rate increasing with $p$) as soon as $p\geq3$. This classical example illustrates that the asymptotic properties of a random walk provides some information on the structure of its state space. This paper aims to explore analogous questions on space made up of combinatorial objects with no algebraic structure. We take as a model for this problem the space of unordered unlabeled rooted trees endowed with Zhang edit distance. To this end, it defines the canonical unbiased random walk on the space of trees and provides an efficient algorithm to evaluate its escape rate. Compared to Zhang algorithm, it is incremental and computes the edit distance along the random walk approximately 100 times faster on trees of size $500$ on average. The escape rate of the random walk on trees is precisely estimated using intensive numerical simulations, out of reasonable reach without the incremental algorithm.

翻译：在$\mathbb{Z}^p$上的简单随机游走展现出两种截然不同的行为，具体取决于$p$的取值：当$p\in\{1,2\}$时，游走是常返的；而当$p\geq3$时，游走则呈现逃逸状态（且逃逸速率随$p$增大而增加）。这一经典范例表明，随机游走的渐近性质能揭示其状态空间结构的若干信息。本文旨在探索由无代数结构的组合对象构成的空间中的类似问题。我们以赋有张氏编辑距离的无标记无根无序树空间作为该问题的模型。为此，本文定义了树空间上的规范无偏随机游走，并提出了一个用于评估其逃逸速率的高效算法。相较于张氏算法，该算法具有增量特性，在平均规模为500的树上沿随机游走计算编辑距离时，速度提升约100倍。通过密集数值模拟，我们精确估计了树上随机游走的逃逸速率，而若无此增量算法，该结果将难以实现。