"True" self-avoiding walks on general trees

TL;DR

This paper analyzes the asymptotic behavior of a specific self-avoiding walk on infinite trees, revealing a phase transition between recurrence and transience based on the tree's branching-ruin number.

Contribution

It establishes a precise phase transition criterion for true self-avoiding walks on trees, linking recurrence and transience to the branching-ruin number.

Findings

Walk is almost surely transient if branching-ruin number > 1/2

Walk is recurrent if branching-ruin number < 1/2

Determines the critical boundary for recurrence and transience

Abstract

We study the asymptotic behavior of ``true" self-avoiding random walks on general infinite locally finite trees. In this model, the walk starts at the root and, at each step, from its current vertex chooses a neighboring edge to traverse with probability proportional to the current weight of that edge, where the weight of each edge after being traversed $n$ times is given by $w(n)=\exp(-\beta n)$. We show that the process exhibits a sharp phase transition between recurrence and transience. The critical value is determined by the branching-ruin number of the tree, which coincides with the Hausdorff dimension of the boundary of the tree under a suitable metric. We prove that the walk is almost surely transient when the branching-ruin number is greater than $1/2$, and recurrent when it is less than $1/2$. This resolves an open question posed by Kosygina.