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Specifically, let $d_N = (1-\\gamma)\\sqrt{\\frac{\\log{N}}{2\\log{\\log{N}}}}$, where $\\gamma \\in (0,1)$ is any fixed real number. We show that for any $\\varepsilon > 0$, any family of groups $G_k$ of order $N_k$ for which $N_k \\to \\infty$, and any integer $2 \\leqslant d\\leqslant d"},"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":"2605.29457","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-05-28T06:49:39Z","cross_cats_sorted":[],"title_canon_sha256":"926dd15fb2ff2b2cac851b9c7df09e379a5dc2bd6fbdd316e059bb05eb1face7","abstract_canon_sha256":"ae1082a326e135be10ca4dcf7ac4a83939ae2a49eb9afa79d48e6b6c263317ff"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-29T01:05:40.495585Z","signature_b64":"hkJZcQWq6QaR86r4T8SHupZvQTBctPy2saNk6W9/SpFJuRYScXJ52yqvrnSAePBCfA13NRmurltbnRW8KPdrDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"91aea0a9a95969fe4f4cd9ffe899a72471a34fc48344dffb68a848051a03d531","last_reissued_at":"2026-05-29T01:05:40.493740Z","signature_status":"signed_v1","first_computed_at":"2026-05-29T01:05:40.493740Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Diameter Thresholds of Random Cayley Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Christina Savvidou, Demetres Christofides, Klas Markstr\\\"om","submitted_at":"2026-05-28T06:49:39Z","abstract_excerpt":"Given a group $G$, the model $\\mathcal{G}(G,p)$ denotes the probability space of all Cayley graphs of $G$ where each element of $G$ is included in the generating set independently at random with probability $p$.\n  In this article, we investigate the threshold probabilities for the diameter of random graphs in this model. Specifically, let $d_N = (1-\\gamma)\\sqrt{\\frac{\\log{N}}{2\\log{\\log{N}}}}$, where $\\gamma \\in (0,1)$ is any fixed real number. We show that for any $\\varepsilon > 0$, any family of groups $G_k$ of order $N_k$ for which $N_k \\to \\infty$, and any integer $2 \\leqslant d\\leqslant d"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.29457","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.29457/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":"2605.29457","created_at":"2026-05-29T01:05:40.494711+00:00"},{"alias_kind":"arxiv_version","alias_value":"2605.29457v1","created_at":"2026-05-29T01:05:40.494711+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.29457","created_at":"2026-05-29T01:05:40.494711+00:00"},{"alias_kind":"pith_short_12","alias_value":"SGXKBKNJLFU7","created_at":"2026-05-29T01:05:40.494711+00:00"},{"alias_kind":"pith_short_16","alias_value":"SGXKBKNJLFU74T2M","created_at":"2026-05-29T01:05:40.494711+00:00"},{"alias_kind":"pith_short_8","alias_value":"SGXKBKNJ","created_at":"2026-05-29T01:05:40.494711+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/SGXKBKNJLFU74T2M3H76RGNHER","json":"https://pith.science/pith/SGXKBKNJLFU74T2M3H76RGNHER.json","graph_json":"https://pith.science/api/pith-number/SGXKBKNJLFU74T2M3H76RGNHER/graph.json","events_json":"https://pith.science/api/pith-number/SGXKBKNJLFU74T2M3H76RGNHER/events.json","paper":"https://pith.science/paper/SGXKBKNJ"},"agent_actions":{"view_html":"https://pith.science/pith/SGXKBKNJLFU74T2M3H76RGNHER","download_json":"https://pith.science/pith/SGXKBKNJLFU74T2M3H76RGNHER.json","view_paper":"https://pith.science/paper/SGXKBKNJ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2605.29457&json=true","fetch_graph":"https://pith.science/api/pith-number/SGXKBKNJLFU74T2M3H76RGNHER/graph.json","fetch_events":"https://pith.science/api/pith-number/SGXKBKNJLFU74T2M3H76RGNHER/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/SGXKBKNJLFU74T2M3H76RGNHER/action/timestamp_anchor","attest_storage":"https://pith.science/pith/SGXKBKNJLFU74T2M3H76RGNHER/action/storage_attestation","attest_author":"https://pith.science/pith/SGXKBKNJLFU74T2M3H76RGNHER/action/author_attestation","sign_citation":"https://pith.science/pith/SGXKBKNJLFU74T2M3H76RGNHER/action/citation_signature","submit_replication":"https://pith.science/pith/SGXKBKNJLFU74T2M3H76RGNHER/action/replication_record"}},"created_at":"2026-05-29T01:05:40.494711+00:00","updated_at":"2026-05-29T01:05:40.494711+00:00"}