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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}=\\"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2606.05830","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2026-06-04T08:08:36Z","cross_cats_sorted":[],"title_canon_sha256":"5c24bbbe39246378c43639e1249156fb6cfaef6c3ce4086d161d5086eb02e1ef","abstract_canon_sha256":"627ad85452fbcff799a654ff178b5a9ef614d1d3eec7e3395296e4f5f7c31efb"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-05T01:15:04.877603Z","signature_b64":"td6Zpwh5LQVyHCItfk4FwEeCYm+eiZdC8A7ZR4RI57wPjk+PGOI/wV5pdf5VOJw5tVAHsV5kCdvrVAafI46WBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"01a5323ceb594366de88fc4bc71a166db90b26c8ff8c6eb1f596858c77322ea8","last_reissued_at":"2026-06-05T01:15:04.877183Z","signature_status":"signed_v1","first_computed_at":"2026-06-05T01:15:04.877183Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $0<\\lambda<1$. 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