Transition probabilities of step-reinforced random walks

TL;DR

This paper introduces a generalized step-reinforced random walk model on groups, providing bounds on transition probabilities and analyzing transience across different spaces, unifying and extending existing models.

Contribution

It develops a unified framework for generalized SRRW on groups, linking transition probability decay to group geometry and resolving open problems.

Findings

Walks are transient in Euclidean space for all dimensions d ≥ 3 and any α<1.

Upper bounds on transition probabilities depend on the isoperimetric profile of the Cayley graph.

Resolves an open problem on exponential decay of the elephant random walk on Cayley trees.

Abstract

The step-reinforced random walk (SRRW), where each step may replicate a randomly chosen past step, exhibits complex dependencies on the history. This paper introduces a generalized SRRW on groups, incorporating arbitrary transformations of past steps, which unifies several existing models in the literature. We develop a unified framework for establishing upper bounds on its transition probabilities for any reinforcement parameter $\alpha<1$, linking the decay rate directly to the geometry of the underlying group. We prove that on Euclidean space, the walk is transient in all dimensions $d \geq 3$ for any $\alpha<1$. On finitely generated groups, we derive the upper bounds using the isoperimetric profile of the Cayley graph, which in particular resolves an open problem regarding the exponential decay of the elephant random walk on Cayley trees.