{"paper":{"title":"Biased Random Walk on $\\mathbb Z_+$ with Traps of Linearly Increasing Depth","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Hua-Ming Wang, Ning Wang","submitted_at":"2026-06-04T08:08:36Z","abstract_excerpt":"We study a $\\lambda$-biased random walk $(X_n)_{n\\ge0}$ on the deterministic infinite rooted tree $\\mathcal{T}=\\{(i,j): i\\ge0,\\,0\\le j\\le i\\}$, whose backbone is $\\{(i,0):i\\ge0\\}$ and, for each $i\\ge1$, the segment $\\{(i,j):1\\le j\\le i\\}$ forms a trap attached to $(i,0)$. The trapping effect induces long sojourns, yielding asymptotics markedly different from simple random walks. The walk is recurrent for $\\lambda\\ge1$ and transient for $0<\\lambda<1$. In the transient regime it is sub-ballistic: its distance from the root grows logarithmically, with \\[ \\liminf_{n\\to\\infty}\\frac{|X_n|}{\\log n}=\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.05830","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.05830/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}