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Sch\\\"utz, Serguei Popov","submitted_at":"2012-10-06T15:43:34Z","abstract_excerpt":"We consider a continuous time random walk $X$ in random environment on $\\Z^+$ such that its potential can be approximated by the function $V: \\R^+\\to \\R$ given by $V(x)=\\sig W(x) -\\frac{b}{1-\\alf}x^{1-\\alf}$ where $\\sig W$ a Brownian motion with diffusion coefficient $\\sig>0$ and parameters $b$, $\\alf$ are such that $b>0$ and $0<\\alf<1/2$. We show that $\\P$-a.s.\\ (where $\\P$ is the averaged law) $\\lim_{t\\to \\infty} \\frac{X_t}{(C^*(\\ln\\ln t)^{-1}\\ln t)^{\\frac{1}{\\alf}}}=1$ with $C^*=\\frac{2\\alf b}{\\sig^2(1-2\\alf)}$. In fact, we prove that by showing that there is a trap located around $(C^*(\\ln"},"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":"1210.1972","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-10-06T15:43:34Z","cross_cats_sorted":[],"title_canon_sha256":"a505ea3ce5f342f3f7febe5a84e11439af776c9da08165fe7df1fba12c7cf6a4","abstract_canon_sha256":"2d5d8e2e71f3fb1f8909e96e459049d170388df41cc544d1b6c3ae20349c128a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:20:58.702626Z","signature_b64":"ha5RnJ34oJyzc2WfGfUsUgXRur29mUl14tPWeBIESwgDxK1+//tEVwuG2IlnP8CPpmK39hc/EhO76N9zmPLcCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"15de42bac68e1a4112395a11470819653e180545983be1d2d20b007afbab08fb","last_reissued_at":"2026-05-18T03:20:58.701853Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:20:58.701853Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Localization for a random walk in slowly decreasing random potential","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Christophe Gallesco, Gunter M. Sch\\\"utz, Serguei Popov","submitted_at":"2012-10-06T15:43:34Z","abstract_excerpt":"We consider a continuous time random walk $X$ in random environment on $\\Z^+$ such that its potential can be approximated by the function $V: \\R^+\\to \\R$ given by $V(x)=\\sig W(x) -\\frac{b}{1-\\alf}x^{1-\\alf}$ where $\\sig W$ a Brownian motion with diffusion coefficient $\\sig>0$ and parameters $b$, $\\alf$ are such that $b>0$ and $0<\\alf<1/2$. We show that $\\P$-a.s.\\ (where $\\P$ is the averaged law) $\\lim_{t\\to \\infty} \\frac{X_t}{(C^*(\\ln\\ln t)^{-1}\\ln t)^{\\frac{1}{\\alf}}}=1$ with $C^*=\\frac{2\\alf b}{\\sig^2(1-2\\alf)}$. In fact, we prove that by showing that there is a trap located around $(C^*(\\ln"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1210.1972","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1210.1972","created_at":"2026-05-18T03:20:58.701985+00:00"},{"alias_kind":"arxiv_version","alias_value":"1210.1972v2","created_at":"2026-05-18T03:20:58.701985+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1210.1972","created_at":"2026-05-18T03:20:58.701985+00:00"},{"alias_kind":"pith_short_12","alias_value":"CXPEFOWGRYNE","created_at":"2026-05-18T12:27:01.376967+00:00"},{"alias_kind":"pith_short_16","alias_value":"CXPEFOWGRYNECERZ","created_at":"2026-05-18T12:27:01.376967+00:00"},{"alias_kind":"pith_short_8","alias_value":"CXPEFOWG","created_at":"2026-05-18T12:27:01.376967+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/CXPEFOWGRYNECERZLIIUOCAZMU","json":"https://pith.science/pith/CXPEFOWGRYNECERZLIIUOCAZMU.json","graph_json":"https://pith.science/api/pith-number/CXPEFOWGRYNECERZLIIUOCAZMU/graph.json","events_json":"https://pith.science/api/pith-number/CXPEFOWGRYNECERZLIIUOCAZMU/events.json","paper":"https://pith.science/paper/CXPEFOWG"},"agent_actions":{"view_html":"https://pith.science/pith/CXPEFOWGRYNECERZLIIUOCAZMU","download_json":"https://pith.science/pith/CXPEFOWGRYNECERZLIIUOCAZMU.json","view_paper":"https://pith.science/paper/CXPEFOWG","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1210.1972&json=true","fetch_graph":"https://pith.science/api/pith-number/CXPEFOWGRYNECERZLIIUOCAZMU/graph.json","fetch_events":"https://pith.science/api/pith-number/CXPEFOWGRYNECERZLIIUOCAZMU/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/CXPEFOWGRYNECERZLIIUOCAZMU/action/timestamp_anchor","attest_storage":"https://pith.science/pith/CXPEFOWGRYNECERZLIIUOCAZMU/action/storage_attestation","attest_author":"https://pith.science/pith/CXPEFOWGRYNECERZLIIUOCAZMU/action/author_attestation","sign_citation":"https://pith.science/pith/CXPEFOWGRYNECERZLIIUOCAZMU/action/citation_signature","submit_replication":"https://pith.science/pith/CXPEFOWGRYNECERZLIIUOCAZMU/action/replication_record"}},"created_at":"2026-05-18T03:20:58.701985+00:00","updated_at":"2026-05-18T03:20:58.701985+00:00"}