# Phase transition for the once-excited random walk on general trees

## TL;DR

This paper investigates the phase transition and recurrence properties of once-excited and multi-excited random walks on general trees, extending previous work on $M$-digging random walks and exploring conditions for recurrence and transience.

## Contribution

It analyzes the critical behavior of once-excited random walks on superperiodic trees and establishes transience results for multi-excited walks on trees with branching number greater than one.

## Key findings

- Critical $M$-digging random walk on superperiodic trees is recurrent.
- Multi-excited random walk on trees with branching number > 1 is transient.
- Techniques from previous studies are effectively applied to new classes of excited random walks.

## Abstract

The phase transition of $M$-digging random on a general tree was studied by Collevecchio, Huynh and Kious (2018). In this paper, we study particularly the critical $M$-digging random walk on a superperiodic tree that is proved to be recurrent. We keep using the techniques introduced by Collevecchio, Kious and Sidoravicius (2017) with the aim of investigating the phase transition of Once-excited random walk on general trees. In addition, we prove if $\mathcal T$ is a tree whose branching number is larger than $1$, any multi-excited random walk on $\mathcal{T}$ moving, after excitation, like a simple random walk is transient.