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.
From: Tuan-Minh Nguyen [view email]
[v1]
Mon, 27 Apr 2026 12:18:08 UTC (42 KB)
[v2]
Thu, 4 Jun 2026 12:27:48 UTC (41 KB)
[v3]
Sun, 2 Aug 2026 01:02:58 UTC (39 KB)
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