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Here $\\phi(x,b)= \\mathbb{P}(\\text{L\\'evy}(1/2,\\kappa(x,b))<0)$ for $\\kappa(x,b) =\n  \\frac{\\sqrt{1-x} b - \\sqrt{1+x}}{\\sqrt{1-x} b + \\sqrt{1+x}}$ and $b=b_S \\geq 0$ measuring the asymptotic asymmetry between positive and negative excursions of the walk (with $b_s=1$ for symmetric increments)."},"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":"1703.10306","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/publicdomain/zero/1.0/","primary_cat":"math.PR","submitted_at":"2017-03-30T04:02:45Z","cross_cats_sorted":[],"title_canon_sha256":"26f6501dc702d0a16874a6a46ed50aafb4c01393cc8b2260a12abdf58414a7f3","abstract_canon_sha256":"614a73b4983f269b30292444e19401ad924cacaf0aaec42e8c0af8a6afa22e13"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:47:37.400423Z","signature_b64":"cilRBsjK0s8Y8kTVFF6F4naI/MSPQr0m3sjDFr9ekA+Y6C2dsKRWqtB6Ew3mJh4ZjP6pEQT4TzJ03GOVx1RsBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"421d7654e4ab7b39abc033669992337ad6a507dad39794dcca84b3ea4ad0a94d","last_reissued_at":"2026-05-18T00:47:37.399794Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:47:37.399794Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Universal Persistence for Local Time of One-dimensional Random Walk","license":"http://creativecommons.org/publicdomain/zero/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Amir Dembo, Jing Miao","submitted_at":"2017-03-30T04:02:45Z","abstract_excerpt":"We prove the power law decay $p(t,x) \\sim t^{-\\phi(x,b)/2}$ in which $p(t,x)$ is the probability that the fraction of time up to $t$ in which a random walk $S$ of i.i.d. zero-mean increments taking finitely many values, is non-negative, exceeds $x$ throughout $s \\in [1,t]$. Here $\\phi(x,b)= \\mathbb{P}(\\text{L\\'evy}(1/2,\\kappa(x,b))<0)$ for $\\kappa(x,b) =\n  \\frac{\\sqrt{1-x} b - \\sqrt{1+x}}{\\sqrt{1-x} b + \\sqrt{1+x}}$ and $b=b_S \\geq 0$ measuring the asymptotic asymmetry between positive and negative excursions of the walk (with $b_s=1$ for symmetric increments)."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1703.10306","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":""},"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":"1703.10306","created_at":"2026-05-18T00:47:37.399889+00:00"},{"alias_kind":"arxiv_version","alias_value":"1703.10306v1","created_at":"2026-05-18T00:47:37.399889+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1703.10306","created_at":"2026-05-18T00:47:37.399889+00:00"},{"alias_kind":"pith_short_12","alias_value":"IIOXMVHEVN5T","created_at":"2026-05-18T12:31:21.493067+00:00"},{"alias_kind":"pith_short_16","alias_value":"IIOXMVHEVN5TTK6A","created_at":"2026-05-18T12:31:21.493067+00:00"},{"alias_kind":"pith_short_8","alias_value":"IIOXMVHE","created_at":"2026-05-18T12:31:21.493067+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/IIOXMVHEVN5TTK6AGNTJTERTPL","json":"https://pith.science/pith/IIOXMVHEVN5TTK6AGNTJTERTPL.json","graph_json":"https://pith.science/api/pith-number/IIOXMVHEVN5TTK6AGNTJTERTPL/graph.json","events_json":"https://pith.science/api/pith-number/IIOXMVHEVN5TTK6AGNTJTERTPL/events.json","paper":"https://pith.science/paper/IIOXMVHE"},"agent_actions":{"view_html":"https://pith.science/pith/IIOXMVHEVN5TTK6AGNTJTERTPL","download_json":"https://pith.science/pith/IIOXMVHEVN5TTK6AGNTJTERTPL.json","view_paper":"https://pith.science/paper/IIOXMVHE","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1703.10306&json=true","fetch_graph":"https://pith.science/api/pith-number/IIOXMVHEVN5TTK6AGNTJTERTPL/graph.json","fetch_events":"https://pith.science/api/pith-number/IIOXMVHEVN5TTK6AGNTJTERTPL/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/IIOXMVHEVN5TTK6AGNTJTERTPL/action/timestamp_anchor","attest_storage":"https://pith.science/pith/IIOXMVHEVN5TTK6AGNTJTERTPL/action/storage_attestation","attest_author":"https://pith.science/pith/IIOXMVHEVN5TTK6AGNTJTERTPL/action/author_attestation","sign_citation":"https://pith.science/pith/IIOXMVHEVN5TTK6AGNTJTERTPL/action/citation_signature","submit_replication":"https://pith.science/pith/IIOXMVHEVN5TTK6AGNTJTERTPL/action/replication_record"}},"created_at":"2026-05-18T00:47:37.399889+00:00","updated_at":"2026-05-18T00:47:37.399889+00:00"}