{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2014:WXA6R6MRLZAFLBOOLYU4BBXPW7","short_pith_number":"pith:WXA6R6MR","schema_version":"1.0","canonical_sha256":"b5c1e8f9915e405585ce5e29c086efb7c518bb4f07a5b96ccdeec2172202f575","source":{"kind":"arxiv","id":"1409.0360","version":3},"attestation_state":"computed","paper":{"title":"Completing the picture for the smallest eigenvalue of real Wishart matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cond-mat.stat-mech","hep-lat","math.MP"],"primary_cat":"math-ph","authors_text":"G. Akemann, M. Kieburg, R. Wegner, T. Guhr, T. Wirtz","submitted_at":"2014-09-01T11:00:10Z","abstract_excerpt":"Rectangular real $N \\times (N + \\nu)$ matrices $W$ with a Gaussian distribution appear very frequently in data analysis, condensed matter physics and quantum field theory. A central question concerns the correlations encoded in the spectral statistics of $WW^T$. The extreme eigenvalues of $W W^T$ are of particular interest. We explicitly compute the distribution and the gap probability of the smallest non-zero eigenvalue in this ensemble, both for arbitrary fixed $N$ and $\\nu$, and in the universal large $N$ limit with $\\nu$ fixed. We uncover an integrable Pfaffian structure valid for all even"},"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":"1409.0360","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2014-09-01T11:00:10Z","cross_cats_sorted":["cond-mat.stat-mech","hep-lat","math.MP"],"title_canon_sha256":"6256bca14feefbaa6a6e91d682fa6485219a2e61cdb398690258eafbb71bd969","abstract_canon_sha256":"13d0346f4f44d670474493cf65ef48ad455d57bf34aa4264bafd42c0fe16a33f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:25:26.356555Z","signature_b64":"NkfChPucfApKe7ycicwi+u3S9F9LDL5elSkE8X8lw7tt2gYM0R4I4rzgd35gAh1IdJU/EhXqSkUxVuMJSjrYAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b5c1e8f9915e405585ce5e29c086efb7c518bb4f07a5b96ccdeec2172202f575","last_reissued_at":"2026-05-18T02:25:26.356167Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:25:26.356167Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Completing the picture for the smallest eigenvalue of real Wishart matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cond-mat.stat-mech","hep-lat","math.MP"],"primary_cat":"math-ph","authors_text":"G. Akemann, M. Kieburg, R. Wegner, T. Guhr, T. Wirtz","submitted_at":"2014-09-01T11:00:10Z","abstract_excerpt":"Rectangular real $N \\times (N + \\nu)$ matrices $W$ with a Gaussian distribution appear very frequently in data analysis, condensed matter physics and quantum field theory. A central question concerns the correlations encoded in the spectral statistics of $WW^T$. The extreme eigenvalues of $W W^T$ are of particular interest. We explicitly compute the distribution and the gap probability of the smallest non-zero eigenvalue in this ensemble, both for arbitrary fixed $N$ and $\\nu$, and in the universal large $N$ limit with $\\nu$ fixed. We uncover an integrable Pfaffian structure valid for all even"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1409.0360","kind":"arxiv","version":3},"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":"1409.0360","created_at":"2026-05-18T02:25:26.356224+00:00"},{"alias_kind":"arxiv_version","alias_value":"1409.0360v3","created_at":"2026-05-18T02:25:26.356224+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1409.0360","created_at":"2026-05-18T02:25:26.356224+00:00"},{"alias_kind":"pith_short_12","alias_value":"WXA6R6MRLZAF","created_at":"2026-05-18T12:28:54.890064+00:00"},{"alias_kind":"pith_short_16","alias_value":"WXA6R6MRLZAFLBOO","created_at":"2026-05-18T12:28:54.890064+00:00"},{"alias_kind":"pith_short_8","alias_value":"WXA6R6MR","created_at":"2026-05-18T12:28:54.890064+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/WXA6R6MRLZAFLBOOLYU4BBXPW7","json":"https://pith.science/pith/WXA6R6MRLZAFLBOOLYU4BBXPW7.json","graph_json":"https://pith.science/api/pith-number/WXA6R6MRLZAFLBOOLYU4BBXPW7/graph.json","events_json":"https://pith.science/api/pith-number/WXA6R6MRLZAFLBOOLYU4BBXPW7/events.json","paper":"https://pith.science/paper/WXA6R6MR"},"agent_actions":{"view_html":"https://pith.science/pith/WXA6R6MRLZAFLBOOLYU4BBXPW7","download_json":"https://pith.science/pith/WXA6R6MRLZAFLBOOLYU4BBXPW7.json","view_paper":"https://pith.science/paper/WXA6R6MR","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1409.0360&json=true","fetch_graph":"https://pith.science/api/pith-number/WXA6R6MRLZAFLBOOLYU4BBXPW7/graph.json","fetch_events":"https://pith.science/api/pith-number/WXA6R6MRLZAFLBOOLYU4BBXPW7/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/WXA6R6MRLZAFLBOOLYU4BBXPW7/action/timestamp_anchor","attest_storage":"https://pith.science/pith/WXA6R6MRLZAFLBOOLYU4BBXPW7/action/storage_attestation","attest_author":"https://pith.science/pith/WXA6R6MRLZAFLBOOLYU4BBXPW7/action/author_attestation","sign_citation":"https://pith.science/pith/WXA6R6MRLZAFLBOOLYU4BBXPW7/action/citation_signature","submit_replication":"https://pith.science/pith/WXA6R6MRLZAFLBOOLYU4BBXPW7/action/replication_record"}},"created_at":"2026-05-18T02:25:26.356224+00:00","updated_at":"2026-05-18T02:25:26.356224+00:00"}