"True" self-avoiding walks on general trees
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866917451939184640 |
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| author | Nguyen, Tuan-Minh |
| author_facet | Nguyen, Tuan-Minh |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_24389 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | "True" self-avoiding walks on general trees Nguyen, Tuan-Minh Probability Statistical Mechanics Mathematical Physics 60K35, 60K37, 82D30 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. |
| title | "True" self-avoiding walks on general trees |
| topic | Probability Statistical Mechanics Mathematical Physics 60K35, 60K37, 82D30 |
| url | https://arxiv.org/abs/2604.24389 |