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We show that the critical probability $p_c(G_r)$ for Bernoulli bond percolation on $G_r$ satisfies $p_c(G_r) \\sim 1/\\mathrm{deg}(G_r)$ as $r\\to\\infty$. This extends work of Penrose and Bollob\\'as-Janson-Riordan, who considered the case $G=\\mathbb{Z}^d$.\n  Our result provides an important ingredient in parallel work of Georgakopoulos in which he introduces "},"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":"2404.17262","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2024-04-26T09:04:01Z","cross_cats_sorted":["math.CO","math.GR"],"title_canon_sha256":"488722d95b7264aab91bf499b174a664cf1e161a7fdd44f7ddef4bf3bee35bd8","abstract_canon_sha256":"bc3c45230553edf3a1646476785f1c4f6e24f1e2feab89940179b1201090125f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T10:27:09.617148Z","signature_b64":"4JMeCBLXpLUbJusmp6194CNKIkk0g12YTp2O8qzCt68KP83gK7siUQLoQQWVx7QM39WsSfMPLJAqDQ5LB0BRBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"675885926937766dd5874c0db43bfd6e0d9e691eddc24b109e39fc8d1df7d0a0","last_reissued_at":"2026-07-05T10:27:09.616155Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T10:27:09.616155Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Spread-out percolation on transitive graphs of polynomial growth","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.GR"],"primary_cat":"math.PR","authors_text":"Matthew Tointon, Panagiotis Spanos","submitted_at":"2024-04-26T09:04:01Z","abstract_excerpt":"Let $G$ be a vertex-transitive graph of superlinear polynomial growth. Given $r>0$, let $G_r$ be the graph on the same vertex set as $G$, with two vertices joined by an edge if and only if they are at graph distance at most $r$ apart in $G$. We show that the critical probability $p_c(G_r)$ for Bernoulli bond percolation on $G_r$ satisfies $p_c(G_r) \\sim 1/\\mathrm{deg}(G_r)$ as $r\\to\\infty$. This extends work of Penrose and Bollob\\'as-Janson-Riordan, who considered the case $G=\\mathbb{Z}^d$.\n  Our result provides an important ingredient in parallel work of Georgakopoulos in which he introduces "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2404.17262","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/2404.17262/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":"2404.17262","created_at":"2026-07-05T10:27:09.616247+00:00"},{"alias_kind":"arxiv_version","alias_value":"2404.17262v2","created_at":"2026-07-05T10:27:09.616247+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2404.17262","created_at":"2026-07-05T10:27:09.616247+00:00"},{"alias_kind":"pith_short_12","alias_value":"M5MILETJG53G","created_at":"2026-07-05T10:27:09.616247+00:00"},{"alias_kind":"pith_short_16","alias_value":"M5MILETJG53G3VMH","created_at":"2026-07-05T10:27:09.616247+00:00"},{"alias_kind":"pith_short_8","alias_value":"M5MILETJ","created_at":"2026-07-05T10:27:09.616247+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/M5MILETJG53G3VMHJQG3IO75NY","json":"https://pith.science/pith/M5MILETJG53G3VMHJQG3IO75NY.json","graph_json":"https://pith.science/api/pith-number/M5MILETJG53G3VMHJQG3IO75NY/graph.json","events_json":"https://pith.science/api/pith-number/M5MILETJG53G3VMHJQG3IO75NY/events.json","paper":"https://pith.science/paper/M5MILETJ"},"agent_actions":{"view_html":"https://pith.science/pith/M5MILETJG53G3VMHJQG3IO75NY","download_json":"https://pith.science/pith/M5MILETJG53G3VMHJQG3IO75NY.json","view_paper":"https://pith.science/paper/M5MILETJ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2404.17262&json=true","fetch_graph":"https://pith.science/api/pith-number/M5MILETJG53G3VMHJQG3IO75NY/graph.json","fetch_events":"https://pith.science/api/pith-number/M5MILETJG53G3VMHJQG3IO75NY/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/M5MILETJG53G3VMHJQG3IO75NY/action/timestamp_anchor","attest_storage":"https://pith.science/pith/M5MILETJG53G3VMHJQG3IO75NY/action/storage_attestation","attest_author":"https://pith.science/pith/M5MILETJG53G3VMHJQG3IO75NY/action/author_attestation","sign_citation":"https://pith.science/pith/M5MILETJG53G3VMHJQG3IO75NY/action/citation_signature","submit_replication":"https://pith.science/pith/M5MILETJG53G3VMHJQG3IO75NY/action/replication_record"}},"created_at":"2026-07-05T10:27:09.616247+00:00","updated_at":"2026-07-05T10:27:09.616247+00:00"}