{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2017:OM2S4INWZEYG3A5JXHFP63YD2K","short_pith_number":"pith:OM2S4INW","schema_version":"1.0","canonical_sha256":"73352e21b6c9306d83a9b9caff6f03d2bfb3b9772093a2626eb03bc41c19d228","source":{"kind":"arxiv","id":"1710.07002","version":2},"attestation_state":"computed","paper":{"title":"Delocalization and Limiting Spectral Distribution of Erd\\H{o}s-R\\'{e}nyi Graphs with Constant Expected Degree","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Jaehun Lee, Paul Jung","submitted_at":"2017-10-19T05:14:14Z","abstract_excerpt":"We consider Erd\\H{o}s-R\\'{e}nyi graphs $G(n,p_n)$ with large constant expected degree $\\lambda$ and $p_n=\\lambda/n$. Bordenave and Lelarge (2010) showed that the infinite-volume limit, in the Benjamini-Schramm topology, is a Galton-Watson tree with offspring distribution Pois($\\lambda$) and the mean spectrum at the root of this tree has unbounded support and corresponds to the limiting spectral distribution of $G(n,p_n)$ as $n\\to\\infty$. We show that if one weights the edges by $1/\\sqrt{\\lambda}$ and sends $\\lambda\\to\\infty$, then the support mostly vanishes and in fact, the limiting spectral "},"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":"1710.07002","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2017-10-19T05:14:14Z","cross_cats_sorted":[],"title_canon_sha256":"3e4454dc495d0c67bc1da9e0997893be607be7deb5a2d45cad04dfb352c96d0c","abstract_canon_sha256":"9689bb48cfd346738a5724577801776cb79edb23ffba7cfc784c10470be21117"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T00:32:35.796441Z","signature_b64":"UEXS2oOl9uqKkgPgCB5T0lAvyQCKCXe1Fm0kPvrBJlAQz89xiwCsEHmXIXe2as9Ixm7+fm33nDSPvZctXOSuBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"73352e21b6c9306d83a9b9caff6f03d2bfb3b9772093a2626eb03bc41c19d228","last_reissued_at":"2026-07-05T00:32:35.796035Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T00:32:35.796035Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Delocalization and Limiting Spectral Distribution of Erd\\H{o}s-R\\'{e}nyi Graphs with Constant Expected Degree","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Jaehun Lee, Paul Jung","submitted_at":"2017-10-19T05:14:14Z","abstract_excerpt":"We consider Erd\\H{o}s-R\\'{e}nyi graphs $G(n,p_n)$ with large constant expected degree $\\lambda$ and $p_n=\\lambda/n$. Bordenave and Lelarge (2010) showed that the infinite-volume limit, in the Benjamini-Schramm topology, is a Galton-Watson tree with offspring distribution Pois($\\lambda$) and the mean spectrum at the root of this tree has unbounded support and corresponds to the limiting spectral distribution of $G(n,p_n)$ as $n\\to\\infty$. We show that if one weights the edges by $1/\\sqrt{\\lambda}$ and sends $\\lambda\\to\\infty$, then the support mostly vanishes and in fact, the limiting spectral "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1710.07002","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/1710.07002/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":"1710.07002","created_at":"2026-07-05T00:32:35.796091+00:00"},{"alias_kind":"arxiv_version","alias_value":"1710.07002v2","created_at":"2026-07-05T00:32:35.796091+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1710.07002","created_at":"2026-07-05T00:32:35.796091+00:00"},{"alias_kind":"pith_short_12","alias_value":"OM2S4INWZEYG","created_at":"2026-07-05T00:32:35.796091+00:00"},{"alias_kind":"pith_short_16","alias_value":"OM2S4INWZEYG3A5J","created_at":"2026-07-05T00:32:35.796091+00:00"},{"alias_kind":"pith_short_8","alias_value":"OM2S4INW","created_at":"2026-07-05T00:32:35.796091+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/OM2S4INWZEYG3A5JXHFP63YD2K","json":"https://pith.science/pith/OM2S4INWZEYG3A5JXHFP63YD2K.json","graph_json":"https://pith.science/api/pith-number/OM2S4INWZEYG3A5JXHFP63YD2K/graph.json","events_json":"https://pith.science/api/pith-number/OM2S4INWZEYG3A5JXHFP63YD2K/events.json","paper":"https://pith.science/paper/OM2S4INW"},"agent_actions":{"view_html":"https://pith.science/pith/OM2S4INWZEYG3A5JXHFP63YD2K","download_json":"https://pith.science/pith/OM2S4INWZEYG3A5JXHFP63YD2K.json","view_paper":"https://pith.science/paper/OM2S4INW","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1710.07002&json=true","fetch_graph":"https://pith.science/api/pith-number/OM2S4INWZEYG3A5JXHFP63YD2K/graph.json","fetch_events":"https://pith.science/api/pith-number/OM2S4INWZEYG3A5JXHFP63YD2K/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/OM2S4INWZEYG3A5JXHFP63YD2K/action/timestamp_anchor","attest_storage":"https://pith.science/pith/OM2S4INWZEYG3A5JXHFP63YD2K/action/storage_attestation","attest_author":"https://pith.science/pith/OM2S4INWZEYG3A5JXHFP63YD2K/action/author_attestation","sign_citation":"https://pith.science/pith/OM2S4INWZEYG3A5JXHFP63YD2K/action/citation_signature","submit_replication":"https://pith.science/pith/OM2S4INWZEYG3A5JXHFP63YD2K/action/replication_record"}},"created_at":"2026-07-05T00:32:35.796091+00:00","updated_at":"2026-07-05T00:32:35.796091+00:00"}