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Define the $\\alpha$-fold self intersection local time $L_n(\\alpha) := \\sum_{x} l(n,x)^{\\alpha}$, and let $L_n(\\alpha|\\epsilon, d)$ the corresponding quantity for $d$-dimensional simple random walk. Without imposing any moment conditions, we show that the variances of the local times $\\mathop{var}(L_n(\\alpha))$ of any genuinely $d$-dimensional random walk are bounded above by the corresponding characteristics of the simple symmetric random walk in $\\mathbb{Z}^d$, i.e. $\\mathop{var"},"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":"1505.07956","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-05-29T08:17:01Z","cross_cats_sorted":[],"title_canon_sha256":"a6979a0b06d07b274b89b78c70a650ad1773cd3ea7fc56251b28fb61ffe2144e","abstract_canon_sha256":"649f9a90b4be5f2d8564651a823261fc37c924428255dd0e63c46dfe0ddb5b89"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:58:26.593141Z","signature_b64":"IBoWlraad822b1P4+YDZlYQ8uqomaBITDrs115ZlQBIJgyitgR14G+BW71NuFEtvCPpzQuV+d37vI+ekRL4ZBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"01bca8998b20791156a54bfc7e5f282e6fb58fe4232d5e2a97e003905d511030","last_reissued_at":"2026-05-18T01:58:26.592680Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:58:26.592680Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Optimal bounds for self-intersection local times","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"George Deligiannidis, Sergey Utev","submitted_at":"2015-05-29T08:17:01Z","abstract_excerpt":"For a random walk $S_n, n\\geq 0$ in $\\mathbb{Z}^d$, let $l(n,x)$ be its local time at the site $x\\in \\mathbb{Z}^d$. Define the $\\alpha$-fold self intersection local time $L_n(\\alpha) := \\sum_{x} l(n,x)^{\\alpha}$, and let $L_n(\\alpha|\\epsilon, d)$ the corresponding quantity for $d$-dimensional simple random walk. Without imposing any moment conditions, we show that the variances of the local times $\\mathop{var}(L_n(\\alpha))$ of any genuinely $d$-dimensional random walk are bounded above by the corresponding characteristics of the simple symmetric random walk in $\\mathbb{Z}^d$, i.e. $\\mathop{var"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1505.07956","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":"1505.07956","created_at":"2026-05-18T01:58:26.592753+00:00"},{"alias_kind":"arxiv_version","alias_value":"1505.07956v2","created_at":"2026-05-18T01:58:26.592753+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1505.07956","created_at":"2026-05-18T01:58:26.592753+00:00"},{"alias_kind":"pith_short_12","alias_value":"AG6KRGMLEB4R","created_at":"2026-05-18T12:29:10.953037+00:00"},{"alias_kind":"pith_short_16","alias_value":"AG6KRGMLEB4RCVVF","created_at":"2026-05-18T12:29:10.953037+00:00"},{"alias_kind":"pith_short_8","alias_value":"AG6KRGML","created_at":"2026-05-18T12:29:10.953037+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/AG6KRGMLEB4RCVVFJP6H4XZIFZ","json":"https://pith.science/pith/AG6KRGMLEB4RCVVFJP6H4XZIFZ.json","graph_json":"https://pith.science/api/pith-number/AG6KRGMLEB4RCVVFJP6H4XZIFZ/graph.json","events_json":"https://pith.science/api/pith-number/AG6KRGMLEB4RCVVFJP6H4XZIFZ/events.json","paper":"https://pith.science/paper/AG6KRGML"},"agent_actions":{"view_html":"https://pith.science/pith/AG6KRGMLEB4RCVVFJP6H4XZIFZ","download_json":"https://pith.science/pith/AG6KRGMLEB4RCVVFJP6H4XZIFZ.json","view_paper":"https://pith.science/paper/AG6KRGML","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1505.07956&json=true","fetch_graph":"https://pith.science/api/pith-number/AG6KRGMLEB4RCVVFJP6H4XZIFZ/graph.json","fetch_events":"https://pith.science/api/pith-number/AG6KRGMLEB4RCVVFJP6H4XZIFZ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/AG6KRGMLEB4RCVVFJP6H4XZIFZ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/AG6KRGMLEB4RCVVFJP6H4XZIFZ/action/storage_attestation","attest_author":"https://pith.science/pith/AG6KRGMLEB4RCVVFJP6H4XZIFZ/action/author_attestation","sign_citation":"https://pith.science/pith/AG6KRGMLEB4RCVVFJP6H4XZIFZ/action/citation_signature","submit_replication":"https://pith.science/pith/AG6KRGMLEB4RCVVFJP6H4XZIFZ/action/replication_record"}},"created_at":"2026-05-18T01:58:26.592753+00:00","updated_at":"2026-05-18T01:58:26.592753+00:00"}