{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2023:B3I4AAZKMBJ5ECV23XSKWF6CV5","short_pith_number":"pith:B3I4AAZK","schema_version":"1.0","canonical_sha256":"0ed1c0032a6053d20abadde4ab17c2af7e6311914604e3ba08792252b810b6d2","source":{"kind":"arxiv","id":"2311.14352","version":2},"attestation_state":"computed","paper":{"title":"The polynomial growth of the infinite long-range percolation cluster","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Johannes B\\\"aumler","submitted_at":"2023-11-24T08:53:07Z","abstract_excerpt":"We study independent long-range percolation on $\\mathbb{Z}^d$ where the nearest-neighbor edges are always open and the probability that two vertices $x,y$ with $\\|x-y\\|>1$ are connected by an edge is proportional to $\\frac{\\beta}{\\|x-y\\|^s}$, where $\\beta>0$ and $s> 0$ are parameters. We show that the ball of radius $k$ centered at the origin in the graph metric grows polynomially if and only if $s\\geq 2d$. For the critical case $s=2d$, we show that the volume growth exponent is inversely proportional to the distance growth exponent. Furthermore, we provide sharp upper and lower bounds on the "},"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":"2311.14352","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2023-11-24T08:53:07Z","cross_cats_sorted":[],"title_canon_sha256":"6ff74b8e105705ef7b8429541fa0d6f9d338d358ba6283fe16cac0112ad84692","abstract_canon_sha256":"a09f7a630d0f2356c0d7ffd497c4325ce370171d8fe54e9997962194da005c50"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T12:07:52.704506Z","signature_b64":"GXtYK6GJIK3g7Qqa2dI43aRdj8/aqt1+XrzqrEeOjpnfuh1meheEmZMJY4cLGCL+kdTJo1aXOeSvisQuaTh5BA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0ed1c0032a6053d20abadde4ab17c2af7e6311914604e3ba08792252b810b6d2","last_reissued_at":"2026-07-05T12:07:52.703992Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T12:07:52.703992Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The polynomial growth of the infinite long-range percolation cluster","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Johannes B\\\"aumler","submitted_at":"2023-11-24T08:53:07Z","abstract_excerpt":"We study independent long-range percolation on $\\mathbb{Z}^d$ where the nearest-neighbor edges are always open and the probability that two vertices $x,y$ with $\\|x-y\\|>1$ are connected by an edge is proportional to $\\frac{\\beta}{\\|x-y\\|^s}$, where $\\beta>0$ and $s> 0$ are parameters. We show that the ball of radius $k$ centered at the origin in the graph metric grows polynomially if and only if $s\\geq 2d$. For the critical case $s=2d$, we show that the volume growth exponent is inversely proportional to the distance growth exponent. Furthermore, we provide sharp upper and lower bounds on the "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2311.14352","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2311.14352/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2311.14352","created_at":"2026-07-05T12:07:52.704059+00:00"},{"alias_kind":"arxiv_version","alias_value":"2311.14352v2","created_at":"2026-07-05T12:07:52.704059+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2311.14352","created_at":"2026-07-05T12:07:52.704059+00:00"},{"alias_kind":"pith_short_12","alias_value":"B3I4AAZKMBJ5","created_at":"2026-07-05T12:07:52.704059+00:00"},{"alias_kind":"pith_short_16","alias_value":"B3I4AAZKMBJ5ECV2","created_at":"2026-07-05T12:07:52.704059+00:00"},{"alias_kind":"pith_short_8","alias_value":"B3I4AAZK","created_at":"2026-07-05T12:07:52.704059+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2506.10511","citing_title":"Uniqueness and dimension for the geodesic of the critical long-range percolation metric","ref_index":3,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/B3I4AAZKMBJ5ECV23XSKWF6CV5","json":"https://pith.science/pith/B3I4AAZKMBJ5ECV23XSKWF6CV5.json","graph_json":"https://pith.science/api/pith-number/B3I4AAZKMBJ5ECV23XSKWF6CV5/graph.json","events_json":"https://pith.science/api/pith-number/B3I4AAZKMBJ5ECV23XSKWF6CV5/events.json","paper":"https://pith.science/paper/B3I4AAZK"},"agent_actions":{"view_html":"https://pith.science/pith/B3I4AAZKMBJ5ECV23XSKWF6CV5","download_json":"https://pith.science/pith/B3I4AAZKMBJ5ECV23XSKWF6CV5.json","view_paper":"https://pith.science/paper/B3I4AAZK","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2311.14352&json=true","fetch_graph":"https://pith.science/api/pith-number/B3I4AAZKMBJ5ECV23XSKWF6CV5/graph.json","fetch_events":"https://pith.science/api/pith-number/B3I4AAZKMBJ5ECV23XSKWF6CV5/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/B3I4AAZKMBJ5ECV23XSKWF6CV5/action/timestamp_anchor","attest_storage":"https://pith.science/pith/B3I4AAZKMBJ5ECV23XSKWF6CV5/action/storage_attestation","attest_author":"https://pith.science/pith/B3I4AAZKMBJ5ECV23XSKWF6CV5/action/author_attestation","sign_citation":"https://pith.science/pith/B3I4AAZKMBJ5ECV23XSKWF6CV5/action/citation_signature","submit_replication":"https://pith.science/pith/B3I4AAZKMBJ5ECV23XSKWF6CV5/action/replication_record"}},"created_at":"2026-07-05T12:07:52.704059+00:00","updated_at":"2026-07-05T12:07:52.704059+00:00"}