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More precisely, let $C_1<d< c_1 n/\\log^2 n$ and let $\\mathcal{M}_{n,d}$ be the set of all $0/1$-valued square $n\\times n$ matrices such that each row and each column of a matrix $M\\in \\mathcal{M}_{n,d}$ has exactly $d$ ones. Let $M$ be uniformly distributed on $\\mathcal{M}_{n,d}$. Then the smallest singular value $s_{n} (M)$ of $M$ is greater than $c_2 n^{-6}$ with probability at least $1-C_2\\log^2 d/\\sqrt{d}$, where $c_1$, $c_2$, $C_1$, a"},"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":"1707.02635","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2017-07-09T20:40:44Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"cbf6318db4f2b59270337c44c7855b0939481b598adaa64e160c57496e498a4b","abstract_canon_sha256":"1ea4ef6aa245d452b8cd6333fe209306975d81d8a927c2ee1b5d3e909b8f1fdf"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:10:24.428047Z","signature_b64":"AZfjBxA3uOs03vIfwS4o72+FtBPnQvMzHHuL157b2yZsViAWbrfZhv2weeqnnEQFgG4uEmCPsWQjsFXEN7OYDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7c89fdb5ed7d53d01e0288219523a6cc875e1f990482955a5fc22749a75ab581","last_reissued_at":"2026-05-18T00:10:24.427323Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:10:24.427323Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The smallest singular value of a shifted $d$-regular random square matrix","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.PR","authors_text":"Alexander Litvak, Anna Lytova, Konstantin Tikhomirov, Nicole Tomczak-Jaegermann, Pierre Youssef","submitted_at":"2017-07-09T20:40:44Z","abstract_excerpt":"We derive a lower bound on the smallest singular value of a random $d$-regular matrix, that is, the adjacency matrix of a random $d$-regular directed graph. More precisely, let $C_1<d< c_1 n/\\log^2 n$ and let $\\mathcal{M}_{n,d}$ be the set of all $0/1$-valued square $n\\times n$ matrices such that each row and each column of a matrix $M\\in \\mathcal{M}_{n,d}$ has exactly $d$ ones. Let $M$ be uniformly distributed on $\\mathcal{M}_{n,d}$. Then the smallest singular value $s_{n} (M)$ of $M$ is greater than $c_2 n^{-6}$ with probability at least $1-C_2\\log^2 d/\\sqrt{d}$, where $c_1$, $c_2$, $C_1$, a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1707.02635","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":"1707.02635","created_at":"2026-05-18T00:10:24.427432+00:00"},{"alias_kind":"arxiv_version","alias_value":"1707.02635v3","created_at":"2026-05-18T00:10:24.427432+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1707.02635","created_at":"2026-05-18T00:10:24.427432+00:00"},{"alias_kind":"pith_short_12","alias_value":"PSE73NPNPVJ5","created_at":"2026-05-18T12:31:37.085036+00:00"},{"alias_kind":"pith_short_16","alias_value":"PSE73NPNPVJ5AHQC","created_at":"2026-05-18T12:31:37.085036+00:00"},{"alias_kind":"pith_short_8","alias_value":"PSE73NPN","created_at":"2026-05-18T12:31:37.085036+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/PSE73NPNPVJ5AHQCRAQZKI5GZS","json":"https://pith.science/pith/PSE73NPNPVJ5AHQCRAQZKI5GZS.json","graph_json":"https://pith.science/api/pith-number/PSE73NPNPVJ5AHQCRAQZKI5GZS/graph.json","events_json":"https://pith.science/api/pith-number/PSE73NPNPVJ5AHQCRAQZKI5GZS/events.json","paper":"https://pith.science/paper/PSE73NPN"},"agent_actions":{"view_html":"https://pith.science/pith/PSE73NPNPVJ5AHQCRAQZKI5GZS","download_json":"https://pith.science/pith/PSE73NPNPVJ5AHQCRAQZKI5GZS.json","view_paper":"https://pith.science/paper/PSE73NPN","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1707.02635&json=true","fetch_graph":"https://pith.science/api/pith-number/PSE73NPNPVJ5AHQCRAQZKI5GZS/graph.json","fetch_events":"https://pith.science/api/pith-number/PSE73NPNPVJ5AHQCRAQZKI5GZS/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/PSE73NPNPVJ5AHQCRAQZKI5GZS/action/timestamp_anchor","attest_storage":"https://pith.science/pith/PSE73NPNPVJ5AHQCRAQZKI5GZS/action/storage_attestation","attest_author":"https://pith.science/pith/PSE73NPNPVJ5AHQCRAQZKI5GZS/action/author_attestation","sign_citation":"https://pith.science/pith/PSE73NPNPVJ5AHQCRAQZKI5GZS/action/citation_signature","submit_replication":"https://pith.science/pith/PSE73NPNPVJ5AHQCRAQZKI5GZS/action/replication_record"}},"created_at":"2026-05-18T00:10:24.427432+00:00","updated_at":"2026-05-18T00:10:24.427432+00:00"}