{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2014:QNEUJK65TRZEHNLKGKQ652GTAR","short_pith_number":"pith:QNEUJK65","schema_version":"1.0","canonical_sha256":"834944abdd9c7243b56a32a1eee8d3045c21c5ffaf399ec56fa6b74233c12391","source":{"kind":"arxiv","id":"1403.1185","version":1},"attestation_state":"computed","paper":{"title":"Phase transitions in the condition number distribution of Gaussian random matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CC","cs.IT","math-ph","math.IT","math.MP","stat.OT"],"primary_cat":"cond-mat.stat-mech","authors_text":"Eytan Katzav, Isaac P\\'erez Castillo, Pierpaolo Vivo","submitted_at":"2014-03-05T16:42:29Z","abstract_excerpt":"We study the statistics of the condition number $\\kappa=\\lambda_{\\mathrm{max}}/\\lambda_{\\mathrm{min}}$ (the ratio between largest and smallest squared singular values) of $N\\times M$ Gaussian random matrices. Using a Coulomb fluid technique, we derive analytically and for large $N$ the cumulative $\\mathcal{P}[\\kappa<x]$ and tail-cumulative $\\mathcal{P}[\\kappa>x]$ distributions of $\\kappa$. We find that these distributions decay as $\\mathcal{P}[\\kappa<x]\\approx\\exp\\left(-\\beta N^2 \\Phi_{-}(x)\\right)$ and $\\mathcal{P}[\\kappa>x]\\approx\\exp\\left(-\\beta N \\Phi_{+}(x)\\right)$, where $\\beta$ is the D"},"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":"1403.1185","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cond-mat.stat-mech","submitted_at":"2014-03-05T16:42:29Z","cross_cats_sorted":["cs.CC","cs.IT","math-ph","math.IT","math.MP","stat.OT"],"title_canon_sha256":"140ec0043296cc8499c2529c1c2b06db3faccae01be84632386be0cf262b7801","abstract_canon_sha256":"aa60983d45c2ae8c6c30810c3100f12dab7c2cdb9b1601f0fdee489c0142a872"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:44:19.514201Z","signature_b64":"4Wj8Ov5ZfukYJNVnwZ6MNAa6l6nnonMpefZKJpz4kEUUunQiHldIDwYVHIe0qN/QbcN/3O3DGdTxPFvcbkc2Aw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"834944abdd9c7243b56a32a1eee8d3045c21c5ffaf399ec56fa6b74233c12391","last_reissued_at":"2026-05-18T01:44:19.513667Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:44:19.513667Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Phase transitions in the condition number distribution of Gaussian random matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CC","cs.IT","math-ph","math.IT","math.MP","stat.OT"],"primary_cat":"cond-mat.stat-mech","authors_text":"Eytan Katzav, Isaac P\\'erez Castillo, Pierpaolo Vivo","submitted_at":"2014-03-05T16:42:29Z","abstract_excerpt":"We study the statistics of the condition number $\\kappa=\\lambda_{\\mathrm{max}}/\\lambda_{\\mathrm{min}}$ (the ratio between largest and smallest squared singular values) of $N\\times M$ Gaussian random matrices. Using a Coulomb fluid technique, we derive analytically and for large $N$ the cumulative $\\mathcal{P}[\\kappa<x]$ and tail-cumulative $\\mathcal{P}[\\kappa>x]$ distributions of $\\kappa$. We find that these distributions decay as $\\mathcal{P}[\\kappa<x]\\approx\\exp\\left(-\\beta N^2 \\Phi_{-}(x)\\right)$ and $\\mathcal{P}[\\kappa>x]\\approx\\exp\\left(-\\beta N \\Phi_{+}(x)\\right)$, where $\\beta$ is the D"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1403.1185","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":""},"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":"1403.1185","created_at":"2026-05-18T01:44:19.513753+00:00"},{"alias_kind":"arxiv_version","alias_value":"1403.1185v1","created_at":"2026-05-18T01:44:19.513753+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1403.1185","created_at":"2026-05-18T01:44:19.513753+00:00"},{"alias_kind":"pith_short_12","alias_value":"QNEUJK65TRZE","created_at":"2026-05-18T12:28:46.137349+00:00"},{"alias_kind":"pith_short_16","alias_value":"QNEUJK65TRZEHNLK","created_at":"2026-05-18T12:28:46.137349+00:00"},{"alias_kind":"pith_short_8","alias_value":"QNEUJK65","created_at":"2026-05-18T12:28:46.137349+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/QNEUJK65TRZEHNLKGKQ652GTAR","json":"https://pith.science/pith/QNEUJK65TRZEHNLKGKQ652GTAR.json","graph_json":"https://pith.science/api/pith-number/QNEUJK65TRZEHNLKGKQ652GTAR/graph.json","events_json":"https://pith.science/api/pith-number/QNEUJK65TRZEHNLKGKQ652GTAR/events.json","paper":"https://pith.science/paper/QNEUJK65"},"agent_actions":{"view_html":"https://pith.science/pith/QNEUJK65TRZEHNLKGKQ652GTAR","download_json":"https://pith.science/pith/QNEUJK65TRZEHNLKGKQ652GTAR.json","view_paper":"https://pith.science/paper/QNEUJK65","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1403.1185&json=true","fetch_graph":"https://pith.science/api/pith-number/QNEUJK65TRZEHNLKGKQ652GTAR/graph.json","fetch_events":"https://pith.science/api/pith-number/QNEUJK65TRZEHNLKGKQ652GTAR/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/QNEUJK65TRZEHNLKGKQ652GTAR/action/timestamp_anchor","attest_storage":"https://pith.science/pith/QNEUJK65TRZEHNLKGKQ652GTAR/action/storage_attestation","attest_author":"https://pith.science/pith/QNEUJK65TRZEHNLKGKQ652GTAR/action/author_attestation","sign_citation":"https://pith.science/pith/QNEUJK65TRZEHNLKGKQ652GTAR/action/citation_signature","submit_replication":"https://pith.science/pith/QNEUJK65TRZEHNLKGKQ652GTAR/action/replication_record"}},"created_at":"2026-05-18T01:44:19.513753+00:00","updated_at":"2026-05-18T01:44:19.513753+00:00"}