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Via a beautiful new framework for large deviation principles in $G(n,p)$, Chatterjee and Varadhan gave bounds on the replica symmetric phase, the region of $(p,r)$ where the answer is positive. They further showed that for any small enough $p$ there are at least two phase transitions as $r$ varies.\n  We settle th"},"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":"1210.7013","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-10-25T22:27:32Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"1ebdaa82c00254a661127f91f1b25c9e692f9a06dff083565bc1590f946a2dae","abstract_canon_sha256":"00ed20ac7c93ab370fc6b7d11ac75da16889acd51879778d6b057e90ddfba29c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:21:20.120250Z","signature_b64":"veH6RuGhsGo1s5wIK9yPoF+CAXGnj4OJ3FF5wq4VkMoBBT7hXEwSAz7j8ycfjdACYcJsuSjMwxhSCf/MKYOTBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"aa6deb4601dafc55faba30dde2fbeabb393b919ea7e2ce2668f480734760f01c","last_reissued_at":"2026-05-18T01:21:20.119557Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:21:20.119557Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On replica symmetry of large deviations in random graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.PR","authors_text":"Eyal Lubetzky, Yufei Zhao","submitted_at":"2012-10-25T22:27:32Z","abstract_excerpt":"The following question is due to Chatterjee and Varadhan (2011). 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They further showed that for any small enough $p$ there are at least two phase transitions as $r$ varies.\n  We settle th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1210.7013","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":"1210.7013","created_at":"2026-05-18T01:21:20.119673+00:00"},{"alias_kind":"arxiv_version","alias_value":"1210.7013v2","created_at":"2026-05-18T01:21:20.119673+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1210.7013","created_at":"2026-05-18T01:21:20.119673+00:00"},{"alias_kind":"pith_short_12","alias_value":"VJW6WRQB3L6F","created_at":"2026-05-18T12:27:25.539911+00:00"},{"alias_kind":"pith_short_16","alias_value":"VJW6WRQB3L6FL6V2","created_at":"2026-05-18T12:27:25.539911+00:00"},{"alias_kind":"pith_short_8","alias_value":"VJW6WRQB","created_at":"2026-05-18T12:27:25.539911+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/VJW6WRQB3L6FL6V2GDO6F67KXM","json":"https://pith.science/pith/VJW6WRQB3L6FL6V2GDO6F67KXM.json","graph_json":"https://pith.science/api/pith-number/VJW6WRQB3L6FL6V2GDO6F67KXM/graph.json","events_json":"https://pith.science/api/pith-number/VJW6WRQB3L6FL6V2GDO6F67KXM/events.json","paper":"https://pith.science/paper/VJW6WRQB"},"agent_actions":{"view_html":"https://pith.science/pith/VJW6WRQB3L6FL6V2GDO6F67KXM","download_json":"https://pith.science/pith/VJW6WRQB3L6FL6V2GDO6F67KXM.json","view_paper":"https://pith.science/paper/VJW6WRQB","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1210.7013&json=true","fetch_graph":"https://pith.science/api/pith-number/VJW6WRQB3L6FL6V2GDO6F67KXM/graph.json","fetch_events":"https://pith.science/api/pith-number/VJW6WRQB3L6FL6V2GDO6F67KXM/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/VJW6WRQB3L6FL6V2GDO6F67KXM/action/timestamp_anchor","attest_storage":"https://pith.science/pith/VJW6WRQB3L6FL6V2GDO6F67KXM/action/storage_attestation","attest_author":"https://pith.science/pith/VJW6WRQB3L6FL6V2GDO6F67KXM/action/author_attestation","sign_citation":"https://pith.science/pith/VJW6WRQB3L6FL6V2GDO6F67KXM/action/citation_signature","submit_replication":"https://pith.science/pith/VJW6WRQB3L6FL6V2GDO6F67KXM/action/replication_record"}},"created_at":"2026-05-18T01:21:20.119673+00:00","updated_at":"2026-05-18T01:21:20.119673+00:00"}