Recurrence and transience of a multi-excited random walk on a regular tree

TL;DR

This paper investigates a multi-excited random walk on a regular tree, revealing a phase transition between recurrence and transience, and explores how excitation order influences the walk's asymptotic behavior and speed.

Contribution

It introduces a generalized model of excited random walks on trees, demonstrating phase transitions and analyzing the impact of excitation order on walk behavior.

Findings

Existence of a recurrence/transience phase transition.

Asymptotic behavior depends on excitation order.

Speed in the transient regime may be non-monotonic.

Abstract

We study a model of multi-excited random walk on a regular tree which generalizes the models of the once excited random walk and the digging random walk introduced by Volkov (2003). We show the existence of a phase transition of the recurrence/transience property of the walk. In particular, we prove that the asymptotic behavior of the walk depends on the order of the excitations, which contrasts with the one dimensional setting studied by Zerner (2005). We also consider the limiting speed of the walk in the transient regime and conjecture that it is not a monotonic function of the environment.