{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:QJERJ4C6WFDQUHD44TR4BFAG5O","short_pith_number":"pith:QJERJ4C6","canonical_record":{"source":{"id":"1103.1919","kind":"arxiv","version":5},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-03-09T23:39:53Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"d8ef106d1aafc3eb26d18af60c3fc06716625c53e9994d0efcff6740b09be6c4","abstract_canon_sha256":"2b8721b4022378e5bb772426824bf701264cee0cb357d8dd28429dd3ff9d4ed2"},"schema_version":"1.0"},"canonical_sha256":"824914f05eb1470a1c7ce4e3c09406eb9502177096e17adc1e39b22f4de185de","source":{"kind":"arxiv","id":"1103.1919","version":5},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1103.1919","created_at":"2026-05-18T03:18:50Z"},{"alias_kind":"arxiv_version","alias_value":"1103.1919v5","created_at":"2026-05-18T03:18:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1103.1919","created_at":"2026-05-18T03:18:50Z"},{"alias_kind":"pith_short_12","alias_value":"QJERJ4C6WFDQ","created_at":"2026-05-18T12:26:39Z"},{"alias_kind":"pith_short_16","alias_value":"QJERJ4C6WFDQUHD4","created_at":"2026-05-18T12:26:39Z"},{"alias_kind":"pith_short_8","alias_value":"QJERJ4C6","created_at":"2026-05-18T12:26:39Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:QJERJ4C6WFDQUHD44TR4BFAG5O","target":"record","payload":{"canonical_record":{"source":{"id":"1103.1919","kind":"arxiv","version":5},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-03-09T23:39:53Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"d8ef106d1aafc3eb26d18af60c3fc06716625c53e9994d0efcff6740b09be6c4","abstract_canon_sha256":"2b8721b4022378e5bb772426824bf701264cee0cb357d8dd28429dd3ff9d4ed2"},"schema_version":"1.0"},"canonical_sha256":"824914f05eb1470a1c7ce4e3c09406eb9502177096e17adc1e39b22f4de185de","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:18:50.018723Z","signature_b64":"Fq6zSYFgPO22BgKy4mVtASjxd2o0sST2T2LGro3GcFkrIIhCFNkS5IKDflNkG1rF0hZ9ViVx8si0JO868nWjCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"824914f05eb1470a1c7ce4e3c09406eb9502177096e17adc1e39b22f4de185de","last_reissued_at":"2026-05-18T03:18:50.017697Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:18:50.017697Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1103.1919","source_version":5,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:18:50Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"xQj2SNTB7Q5fVrosM3usl9OrfFwUHGRvUIFQ6/AvBZv7TE5QqzJlMRO4CAtoo3jJeTldlTJYZEJNcUkEhG4GDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T05:56:52.007977Z"},"content_sha256":"c1227340450eb9b725ec21a3ea8b44110c77034f6ea490c330856331d07b6cb6","schema_version":"1.0","event_id":"sha256:c1227340450eb9b725ec21a3ea8b44110c77034f6ea490c330856331d07b6cb6"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:QJERJ4C6WFDQUHD44TR4BFAG5O","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Spectral statistics of Erd\\H{o}s-R\\'{e}nyi graphs I: Local semicircle law","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.PR","authors_text":"Antti Knowles, Horng-Tzer Yau, Jun Yin, L\\'aszl\\'o Erd\\H{o}s","submitted_at":"2011-03-09T23:39:53Z","abstract_excerpt":"We consider the ensemble of adjacency matrices of Erd\\H{o}s-R\\'{e}nyi random graphs, that is, graphs on $N$ vertices where every edge is chosen independently and with probability $p\\equiv p(N)$. We rescale the matrix so that its bulk eigenvalues are of order one. We prove that, as long as $pN\\to\\infty$ (with a speed at least logarithmic in $N$), the density of eigenvalues of the Erd\\H{o}s-R\\'{e}nyi ensemble is given by the Wigner semicircle law for spectral windows of length larger than $N^{-1}$ (up to logarithmic corrections). As a consequence, all eigenvectors are proved to be completely del"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1103.1919","kind":"arxiv","version":5},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:18:50Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"xEBHWfKpXFc0O49RpP3dyJ4iKdlIR7oa4U+DC/gKHrFIYxZTpcwHElURIeJyCrRNX18AR0AkaIM5oUsqb8INBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T05:56:52.008596Z"},"content_sha256":"f4f72e1724d8ba02739ca2940c765223d06eb41b248178e7e3b28396a890ce4d","schema_version":"1.0","event_id":"sha256:f4f72e1724d8ba02739ca2940c765223d06eb41b248178e7e3b28396a890ce4d"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/QJERJ4C6WFDQUHD44TR4BFAG5O/bundle.json","state_url":"https://pith.science/pith/QJERJ4C6WFDQUHD44TR4BFAG5O/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/QJERJ4C6WFDQUHD44TR4BFAG5O/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":"2026-05-28T05:56:52Z","links":{"resolver":"https://pith.science/pith/QJERJ4C6WFDQUHD44TR4BFAG5O","bundle":"https://pith.science/pith/QJERJ4C6WFDQUHD44TR4BFAG5O/bundle.json","state":"https://pith.science/pith/QJERJ4C6WFDQUHD44TR4BFAG5O/state.json","well_known_bundle":"https://pith.science/.well-known/pith/QJERJ4C6WFDQUHD44TR4BFAG5O/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:QJERJ4C6WFDQUHD44TR4BFAG5O","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":"2b8721b4022378e5bb772426824bf701264cee0cb357d8dd28429dd3ff9d4ed2","cross_cats_sorted":["math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-03-09T23:39:53Z","title_canon_sha256":"d8ef106d1aafc3eb26d18af60c3fc06716625c53e9994d0efcff6740b09be6c4"},"schema_version":"1.0","source":{"id":"1103.1919","kind":"arxiv","version":5}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1103.1919","created_at":"2026-05-18T03:18:50Z"},{"alias_kind":"arxiv_version","alias_value":"1103.1919v5","created_at":"2026-05-18T03:18:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1103.1919","created_at":"2026-05-18T03:18:50Z"},{"alias_kind":"pith_short_12","alias_value":"QJERJ4C6WFDQ","created_at":"2026-05-18T12:26:39Z"},{"alias_kind":"pith_short_16","alias_value":"QJERJ4C6WFDQUHD4","created_at":"2026-05-18T12:26:39Z"},{"alias_kind":"pith_short_8","alias_value":"QJERJ4C6","created_at":"2026-05-18T12:26:39Z"}],"graph_snapshots":[{"event_id":"sha256:f4f72e1724d8ba02739ca2940c765223d06eb41b248178e7e3b28396a890ce4d","target":"graph","created_at":"2026-05-18T03:18:50Z","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"},"paper":{"abstract_excerpt":"We consider the ensemble of adjacency matrices of Erd\\H{o}s-R\\'{e}nyi random graphs, that is, graphs on $N$ vertices where every edge is chosen independently and with probability $p\\equiv p(N)$. We rescale the matrix so that its bulk eigenvalues are of order one. We prove that, as long as $pN\\to\\infty$ (with a speed at least logarithmic in $N$), the density of eigenvalues of the Erd\\H{o}s-R\\'{e}nyi ensemble is given by the Wigner semicircle law for spectral windows of length larger than $N^{-1}$ (up to logarithmic corrections). As a consequence, all eigenvectors are proved to be completely del","authors_text":"Antti Knowles, Horng-Tzer Yau, Jun Yin, L\\'aszl\\'o Erd\\H{o}s","cross_cats":["math-ph","math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-03-09T23:39:53Z","title":"Spectral statistics of Erd\\H{o}s-R\\'{e}nyi graphs I: Local semicircle law"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1103.1919","kind":"arxiv","version":5},"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:c1227340450eb9b725ec21a3ea8b44110c77034f6ea490c330856331d07b6cb6","target":"record","created_at":"2026-05-18T03:18:50Z","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":"2b8721b4022378e5bb772426824bf701264cee0cb357d8dd28429dd3ff9d4ed2","cross_cats_sorted":["math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-03-09T23:39:53Z","title_canon_sha256":"d8ef106d1aafc3eb26d18af60c3fc06716625c53e9994d0efcff6740b09be6c4"},"schema_version":"1.0","source":{"id":"1103.1919","kind":"arxiv","version":5}},"canonical_sha256":"824914f05eb1470a1c7ce4e3c09406eb9502177096e17adc1e39b22f4de185de","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"824914f05eb1470a1c7ce4e3c09406eb9502177096e17adc1e39b22f4de185de","first_computed_at":"2026-05-18T03:18:50.017697Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:18:50.017697Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Fq6zSYFgPO22BgKy4mVtASjxd2o0sST2T2LGro3GcFkrIIhCFNkS5IKDflNkG1rF0hZ9ViVx8si0JO868nWjCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:18:50.018723Z","signed_message":"canonical_sha256_bytes"},"source_id":"1103.1919","source_kind":"arxiv","source_version":5}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c1227340450eb9b725ec21a3ea8b44110c77034f6ea490c330856331d07b6cb6","sha256:f4f72e1724d8ba02739ca2940c765223d06eb41b248178e7e3b28396a890ce4d"],"state_sha256":"39e89fddb0249c06d6d7ab290d8033000bb9082853d3d5e2b93d12be9dc7d51f"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"NQxDddvbFUQgaa88vghrkfInE8jbvNm1TxW5tREZY6r0XDco6dHj9zH3WsRvLU86OalS+X52AYj0KISm9PeCAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-28T05:56:52.011771Z","bundle_sha256":"d953495242ebc3cc1a88fc65e2e2873c97b3525a243c2d1063f728c0128c0997"}}