{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2023:NANCEFJMPDFROA6RWCQHTAVS5Z","short_pith_number":"pith:NANCEFJM","canonical_record":{"source":{"id":"2309.11007","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2023-09-20T02:23:13Z","cross_cats_sorted":["math-ph","math.MP","math.SP"],"title_canon_sha256":"7db7dd770e642ecf8223329ae5acc3c0c43794da7dac64c2b8d2a101163521f5","abstract_canon_sha256":"8e6c54f99f18be5ea1c44a652821d27f3b8c0f847cc521956482553283da6f7a"},"schema_version":"1.0"},"canonical_sha256":"681a22152c78cb1703d1b0a07982b2ee66f1999bf130513fe48fb51e93cc8e5b","source":{"kind":"arxiv","id":"2309.11007","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2309.11007","created_at":"2026-07-05T06:52:29Z"},{"alias_kind":"arxiv_version","alias_value":"2309.11007v1","created_at":"2026-07-05T06:52:29Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2309.11007","created_at":"2026-07-05T06:52:29Z"},{"alias_kind":"pith_short_12","alias_value":"NANCEFJMPDFR","created_at":"2026-07-05T06:52:29Z"},{"alias_kind":"pith_short_16","alias_value":"NANCEFJMPDFROA6R","created_at":"2026-07-05T06:52:29Z"},{"alias_kind":"pith_short_8","alias_value":"NANCEFJM","created_at":"2026-07-05T06:52:29Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2023:NANCEFJMPDFROA6RWCQHTAVS5Z","target":"record","payload":{"canonical_record":{"source":{"id":"2309.11007","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2023-09-20T02:23:13Z","cross_cats_sorted":["math-ph","math.MP","math.SP"],"title_canon_sha256":"7db7dd770e642ecf8223329ae5acc3c0c43794da7dac64c2b8d2a101163521f5","abstract_canon_sha256":"8e6c54f99f18be5ea1c44a652821d27f3b8c0f847cc521956482553283da6f7a"},"schema_version":"1.0"},"canonical_sha256":"681a22152c78cb1703d1b0a07982b2ee66f1999bf130513fe48fb51e93cc8e5b","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T06:52:29.800816Z","signature_b64":"gXpx8a2ZZ9Y2FgprBJ/CbZsr2SM+4bopdlE6DMbu2eQ8s3CPwq8Ctpi0Gu8T8mhJD3/sZslX8jC5PPh5BCc1Aw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"681a22152c78cb1703d1b0a07982b2ee66f1999bf130513fe48fb51e93cc8e5b","last_reissued_at":"2026-07-05T06:52:29.800383Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T06:52:29.800383Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2309.11007","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T06:52:29Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"oHm4gPxih7nUIwOrmqcXEeLBApeAmohD+EGwBkXYkWVIlC1Xm69imJ5pM4QLkeVi5rt8i2xixYu9KO91V8yjBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-12T05:01:40.730726Z"},"content_sha256":"b82d0e8c8204c4b2421e6e64295ed78ba1fa5c1184058c9f68993735dbd9baaf","schema_version":"1.0","event_id":"sha256:b82d0e8c8204c4b2421e6e64295ed78ba1fa5c1184058c9f68993735dbd9baaf"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2023:NANCEFJMPDFROA6RWCQHTAVS5Z","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The Spectral Edge of Constant Degree Erd\\H{o}s-R\\'{e}nyi Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP","math.SP"],"primary_cat":"math.PR","authors_text":"Ella Hiesmayr, Theo McKenzie","submitted_at":"2023-09-20T02:23:13Z","abstract_excerpt":"We show that for an Erd\\H{o}s-R\\'{e}nyi graph on $N$ vertices with expected degree $d$ satisfying $\\log^{-1/9}N\\leq d\\leq \\log^{1/40}N$, the largest eigenvalues can be precisely determined by small neighborhoods around vertices of close to maximal degree. Moreover, under the added condition that $d\\geq\\log^{-1/15}N$, the corresponding eigenvectors are localized, in that the mass of the eigenvector decays exponentially away from the high degree vertex. This dependence on local neighborhoods implies that the edge eigenvalues converge to a Poisson point process. These theorems extend a result of "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2309.11007","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/2309.11007/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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T06:52:29Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/pFK0uU7gMsR7V/duXUpf8jwXC8HFIQZ/077bhCqdwYpWcWRl5SUb8DSLfUPBMSyIxqSQdgnFji+/FjcJD5kAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-12T05:01:40.731363Z"},"content_sha256":"94b5d1d94815bf8df5f843176f0677d95a028bae72a8b26a2c455619ad947f75","schema_version":"1.0","event_id":"sha256:94b5d1d94815bf8df5f843176f0677d95a028bae72a8b26a2c455619ad947f75"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/NANCEFJMPDFROA6RWCQHTAVS5Z/bundle.json","state_url":"https://pith.science/pith/NANCEFJMPDFROA6RWCQHTAVS5Z/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/NANCEFJMPDFROA6RWCQHTAVS5Z/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-08-12T05:01:40Z","links":{"resolver":"https://pith.science/pith/NANCEFJMPDFROA6RWCQHTAVS5Z","bundle":"https://pith.science/pith/NANCEFJMPDFROA6RWCQHTAVS5Z/bundle.json","state":"https://pith.science/pith/NANCEFJMPDFROA6RWCQHTAVS5Z/state.json","well_known_bundle":"https://pith.science/.well-known/pith/NANCEFJMPDFROA6RWCQHTAVS5Z/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2023:NANCEFJMPDFROA6RWCQHTAVS5Z","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"8e6c54f99f18be5ea1c44a652821d27f3b8c0f847cc521956482553283da6f7a","cross_cats_sorted":["math-ph","math.MP","math.SP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2023-09-20T02:23:13Z","title_canon_sha256":"7db7dd770e642ecf8223329ae5acc3c0c43794da7dac64c2b8d2a101163521f5"},"schema_version":"1.0","source":{"id":"2309.11007","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2309.11007","created_at":"2026-07-05T06:52:29Z"},{"alias_kind":"arxiv_version","alias_value":"2309.11007v1","created_at":"2026-07-05T06:52:29Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2309.11007","created_at":"2026-07-05T06:52:29Z"},{"alias_kind":"pith_short_12","alias_value":"NANCEFJMPDFR","created_at":"2026-07-05T06:52:29Z"},{"alias_kind":"pith_short_16","alias_value":"NANCEFJMPDFROA6R","created_at":"2026-07-05T06:52:29Z"},{"alias_kind":"pith_short_8","alias_value":"NANCEFJM","created_at":"2026-07-05T06:52:29Z"}],"graph_snapshots":[{"event_id":"sha256:94b5d1d94815bf8df5f843176f0677d95a028bae72a8b26a2c455619ad947f75","target":"graph","created_at":"2026-07-05T06:52:29Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2309.11007/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We show that for an Erd\\H{o}s-R\\'{e}nyi graph on $N$ vertices with expected degree $d$ satisfying $\\log^{-1/9}N\\leq d\\leq \\log^{1/40}N$, the largest eigenvalues can be precisely determined by small neighborhoods around vertices of close to maximal degree. Moreover, under the added condition that $d\\geq\\log^{-1/15}N$, the corresponding eigenvectors are localized, in that the mass of the eigenvector decays exponentially away from the high degree vertex. This dependence on local neighborhoods implies that the edge eigenvalues converge to a Poisson point process. These theorems extend a result of ","authors_text":"Ella Hiesmayr, Theo McKenzie","cross_cats":["math-ph","math.MP","math.SP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2023-09-20T02:23:13Z","title":"The Spectral Edge of Constant Degree Erd\\H{o}s-R\\'{e}nyi Graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2309.11007","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:b82d0e8c8204c4b2421e6e64295ed78ba1fa5c1184058c9f68993735dbd9baaf","target":"record","created_at":"2026-07-05T06:52:29Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"8e6c54f99f18be5ea1c44a652821d27f3b8c0f847cc521956482553283da6f7a","cross_cats_sorted":["math-ph","math.MP","math.SP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2023-09-20T02:23:13Z","title_canon_sha256":"7db7dd770e642ecf8223329ae5acc3c0c43794da7dac64c2b8d2a101163521f5"},"schema_version":"1.0","source":{"id":"2309.11007","kind":"arxiv","version":1}},"canonical_sha256":"681a22152c78cb1703d1b0a07982b2ee66f1999bf130513fe48fb51e93cc8e5b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"681a22152c78cb1703d1b0a07982b2ee66f1999bf130513fe48fb51e93cc8e5b","first_computed_at":"2026-07-05T06:52:29.800383Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T06:52:29.800383Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"gXpx8a2ZZ9Y2FgprBJ/CbZsr2SM+4bopdlE6DMbu2eQ8s3CPwq8Ctpi0Gu8T8mhJD3/sZslX8jC5PPh5BCc1Aw==","signature_status":"signed_v1","signed_at":"2026-07-05T06:52:29.800816Z","signed_message":"canonical_sha256_bytes"},"source_id":"2309.11007","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b82d0e8c8204c4b2421e6e64295ed78ba1fa5c1184058c9f68993735dbd9baaf","sha256:94b5d1d94815bf8df5f843176f0677d95a028bae72a8b26a2c455619ad947f75"],"state_sha256":"9bc82e774c56a15eaff388812b8a352d282808d55ee0bd442588fb04bfc7f7b2"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"lEN40DUcbvcTovJ1eT3gYkkYzkj3/p3Z66+4oH8WZXq16bqYvHyhCwf+RXKVNOXw7rDfGnqOle/yhXASEeafDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-12T05:01:40.736825Z","bundle_sha256":"915a93e285c68044b67f65cd87e3bda0f11a9be02976493788ff847f6c1fc6c6"}}