{"paper":{"title":"\"True\" self-avoiding walks on general trees","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"True self-avoiding walks on trees switch from recurrent to transient exactly when the branching-ruin number exceeds 1/2.","cross_cats":["cond-mat.stat-mech","math-ph","math.MP"],"primary_cat":"math.PR","authors_text":"Tuan-Minh Nguyen","submitted_at":"2026-04-27T12:18:08Z","abstract_excerpt":"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 Hausd"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"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. 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.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The branching-ruin number of the tree equals the Hausdorff dimension of its boundary under a suitable metric, allowing the critical value to be located at 1/2.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"True self-avoiding walks on general trees are transient if the branching-ruin number exceeds 1/2 and recurrent otherwise.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"True self-avoiding walks on trees switch from recurrent to transient exactly when the branching-ruin number exceeds 1/2.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"3381a42417d80c5c1eb7460417f897ec9d9fba0f9e4b49e96328e12d6a5c2f14"},"source":{"id":"2604.24389","kind":"arxiv","version":2},"verdict":{"id":"a2fadedf-c578-4338-97d6-4cdf11b4b1ef","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-08T01:55:12.268244Z","strongest_claim":"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. 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.","one_line_summary":"True self-avoiding walks on general trees are transient if the branching-ruin number exceeds 1/2 and recurrent otherwise.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The branching-ruin number of the tree equals the Hausdorff dimension of its boundary under a suitable metric, allowing the critical value to be located at 1/2.","pith_extraction_headline":"True self-avoiding walks on trees switch from recurrent to transient exactly when the branching-ruin number exceeds 1/2."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.24389/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"ai_meta_artifact","ran_at":"2026-05-21T06:41:15.018328Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T22:06:18.380624Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"c327ccac50bcc3992c385e327435301f9ce7588d18a9a0cc8ea45f6018e43e65"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}