"True" self-avoiding walks on general trees

2026/04/30 by Tuan-Minh Nguyen
Mathematics · Physics and Astronomy · #math.PR #cond-mat.stat-mech #math-ph #math.MP #msc:60K35 #msc:60K37 #msc:82D30

paper · pdf

42 pages

arxiv created 2026/08/02 · arxiv updated 2026/08/04

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(-βn). 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.

Citations

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