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For an N-step trajectory X=(X_0,X_1,...,X_N) and a monotone convex function V: R^+ -> R^+ with V(0)=0, define V(X)= sum_{j=1}^{N-1} V(|X_j|). Further, let I_{N,+}^{a,b} be the set of all non-negative paths X compatible with the boundary conditions X_0=a, X_N=b. We discuss asymptotic properties of X in I_{N,+}^{a,b} w.r.t. the probability distribution P_{N}^{a,b}(X)= (Z_{N}^{a,b})^{-1} exp{-la"},"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":"math/0310217","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.PR","submitted_at":"2003-10-15T12:38:16Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"3a61bdfb82b159be505b260e78d98a640a7901fe3ba41e459ecdc37d3acb0c67","abstract_canon_sha256":"57bb4935ca27804dd59dd678118892448a1bfd9303dc42b65a01f302815545a6"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:14:46.414335Z","signature_b64":"Ofr8b/ZPZ6ly9036lG04BfObqOCfzVMedas88nU7uIe4UfsMvGr4FCCBRJKs6kOdKs1xrWVKXzjuwrnBitIlAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c821f51ee7c4957c88058d1c14598990f4334e480d95d4c1db61662efd945bee","last_reissued_at":"2026-05-18T04:14:46.413879Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:14:46.413879Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Universality of Critical Behaviour in a Class of Recurrent Random Walks","license":"","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.PR","authors_text":"Ostap Hryniv, Yvan Velenik","submitted_at":"2003-10-15T12:38:16Z","abstract_excerpt":"Let X_0=0, X_1, X_2, ..., be an aperiodic random walk generated by a sequence xi_1, xi_2, ..., of i.i.d. integer-valued random variables with common distribution p(.) having zero mean and finite variance. For an N-step trajectory X=(X_0,X_1,...,X_N) and a monotone convex function V: R^+ -> R^+ with V(0)=0, define V(X)= sum_{j=1}^{N-1} V(|X_j|). Further, let I_{N,+}^{a,b} be the set of all non-negative paths X compatible with the boundary conditions X_0=a, X_N=b. We discuss asymptotic properties of X in I_{N,+}^{a,b} w.r.t. the probability distribution P_{N}^{a,b}(X)= (Z_{N}^{a,b})^{-1} exp{-la"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0310217","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":"math/0310217","created_at":"2026-05-18T04:14:46.413949+00:00"},{"alias_kind":"arxiv_version","alias_value":"math/0310217v1","created_at":"2026-05-18T04:14:46.413949+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0310217","created_at":"2026-05-18T04:14:46.413949+00:00"},{"alias_kind":"pith_short_12","alias_value":"ZAQ7KHXHYSKX","created_at":"2026-05-18T12:25:52.051335+00:00"},{"alias_kind":"pith_short_16","alias_value":"ZAQ7KHXHYSKXZCAF","created_at":"2026-05-18T12:25:52.051335+00:00"},{"alias_kind":"pith_short_8","alias_value":"ZAQ7KHXH","created_at":"2026-05-18T12:25:52.051335+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/ZAQ7KHXHYSKXZCAFRUOBIWMJSD","json":"https://pith.science/pith/ZAQ7KHXHYSKXZCAFRUOBIWMJSD.json","graph_json":"https://pith.science/api/pith-number/ZAQ7KHXHYSKXZCAFRUOBIWMJSD/graph.json","events_json":"https://pith.science/api/pith-number/ZAQ7KHXHYSKXZCAFRUOBIWMJSD/events.json","paper":"https://pith.science/paper/ZAQ7KHXH"},"agent_actions":{"view_html":"https://pith.science/pith/ZAQ7KHXHYSKXZCAFRUOBIWMJSD","download_json":"https://pith.science/pith/ZAQ7KHXHYSKXZCAFRUOBIWMJSD.json","view_paper":"https://pith.science/paper/ZAQ7KHXH","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=math/0310217&json=true","fetch_graph":"https://pith.science/api/pith-number/ZAQ7KHXHYSKXZCAFRUOBIWMJSD/graph.json","fetch_events":"https://pith.science/api/pith-number/ZAQ7KHXHYSKXZCAFRUOBIWMJSD/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ZAQ7KHXHYSKXZCAFRUOBIWMJSD/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ZAQ7KHXHYSKXZCAFRUOBIWMJSD/action/storage_attestation","attest_author":"https://pith.science/pith/ZAQ7KHXHYSKXZCAFRUOBIWMJSD/action/author_attestation","sign_citation":"https://pith.science/pith/ZAQ7KHXHYSKXZCAFRUOBIWMJSD/action/citation_signature","submit_replication":"https://pith.science/pith/ZAQ7KHXHYSKXZCAFRUOBIWMJSD/action/replication_record"}},"created_at":"2026-05-18T04:14:46.413949+00:00","updated_at":"2026-05-18T04:14:46.413949+00:00"}