{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2013:A5IVZ7FWTFH7C2VSON3V2M4G7P","short_pith_number":"pith:A5IVZ7FW","schema_version":"1.0","canonical_sha256":"07515cfcb6994ff16ab273775d3386fbd2a07d8dcfd5cd723db2d2b3e43d89d7","source":{"kind":"arxiv","id":"1311.6783","version":1},"attestation_state":"computed","paper":{"title":"Local Spectrum of Truncations of Kronecker Products of Haar Distributed Unitary Matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Brendan Farrell, Raj Rao Nadakuditi","submitted_at":"2013-11-26T19:31:12Z","abstract_excerpt":"We address the local spectral behavior of the random matrix $\\Pi_1 U^{\\otimes k} \\Pi_2 U^{\\otimes k *} \\Pi_1$, where $U$ is a Haar distributed unitary matrix of size $n\\times n$, the factor $k$ is at most $c_0\\log n$ for a small constant $c_0>0$, and $\\Pi_1,\\Pi_2$ are arbitrary projections on $\\ell_2^{n^k}$ of ranks proportional to $n^k$. We prove that in this setting the $k$-fold Kronecker product behaves similarly to the well-studied case when $k=1$."},"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":"1311.6783","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-11-26T19:31:12Z","cross_cats_sorted":[],"title_canon_sha256":"e3fed9d61cce8a2d29fcc5e7509183d69168aacb3dc6377fc87e21373c0b5348","abstract_canon_sha256":"2ddc09df792d8ebb9ac4d3be2157012085d6af3f1a529ae13c11f8b09efd3875"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:06:05.835798Z","signature_b64":"0lEu6jxQrVEowVWWDPP2Yv7z0ujJ07J5NxlQCnD/kjAUixbp9TmM4q5hCTQPPWDlO0DaSicjAU/0G2z0nQi+BQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"07515cfcb6994ff16ab273775d3386fbd2a07d8dcfd5cd723db2d2b3e43d89d7","last_reissued_at":"2026-05-18T03:06:05.835349Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:06:05.835349Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Local Spectrum of Truncations of Kronecker Products of Haar Distributed Unitary Matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Brendan Farrell, Raj Rao Nadakuditi","submitted_at":"2013-11-26T19:31:12Z","abstract_excerpt":"We address the local spectral behavior of the random matrix $\\Pi_1 U^{\\otimes k} \\Pi_2 U^{\\otimes k *} \\Pi_1$, where $U$ is a Haar distributed unitary matrix of size $n\\times n$, the factor $k$ is at most $c_0\\log n$ for a small constant $c_0>0$, and $\\Pi_1,\\Pi_2$ are arbitrary projections on $\\ell_2^{n^k}$ of ranks proportional to $n^k$. We prove that in this setting the $k$-fold Kronecker product behaves similarly to the well-studied case when $k=1$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1311.6783","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":"1311.6783","created_at":"2026-05-18T03:06:05.835421+00:00"},{"alias_kind":"arxiv_version","alias_value":"1311.6783v1","created_at":"2026-05-18T03:06:05.835421+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1311.6783","created_at":"2026-05-18T03:06:05.835421+00:00"},{"alias_kind":"pith_short_12","alias_value":"A5IVZ7FWTFH7","created_at":"2026-05-18T12:27:38.830355+00:00"},{"alias_kind":"pith_short_16","alias_value":"A5IVZ7FWTFH7C2VS","created_at":"2026-05-18T12:27:38.830355+00:00"},{"alias_kind":"pith_short_8","alias_value":"A5IVZ7FW","created_at":"2026-05-18T12:27:38.830355+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/A5IVZ7FWTFH7C2VSON3V2M4G7P","json":"https://pith.science/pith/A5IVZ7FWTFH7C2VSON3V2M4G7P.json","graph_json":"https://pith.science/api/pith-number/A5IVZ7FWTFH7C2VSON3V2M4G7P/graph.json","events_json":"https://pith.science/api/pith-number/A5IVZ7FWTFH7C2VSON3V2M4G7P/events.json","paper":"https://pith.science/paper/A5IVZ7FW"},"agent_actions":{"view_html":"https://pith.science/pith/A5IVZ7FWTFH7C2VSON3V2M4G7P","download_json":"https://pith.science/pith/A5IVZ7FWTFH7C2VSON3V2M4G7P.json","view_paper":"https://pith.science/paper/A5IVZ7FW","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1311.6783&json=true","fetch_graph":"https://pith.science/api/pith-number/A5IVZ7FWTFH7C2VSON3V2M4G7P/graph.json","fetch_events":"https://pith.science/api/pith-number/A5IVZ7FWTFH7C2VSON3V2M4G7P/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/A5IVZ7FWTFH7C2VSON3V2M4G7P/action/timestamp_anchor","attest_storage":"https://pith.science/pith/A5IVZ7FWTFH7C2VSON3V2M4G7P/action/storage_attestation","attest_author":"https://pith.science/pith/A5IVZ7FWTFH7C2VSON3V2M4G7P/action/author_attestation","sign_citation":"https://pith.science/pith/A5IVZ7FWTFH7C2VSON3V2M4G7P/action/citation_signature","submit_replication":"https://pith.science/pith/A5IVZ7FWTFH7C2VSON3V2M4G7P/action/replication_record"}},"created_at":"2026-05-18T03:06:05.835421+00:00","updated_at":"2026-05-18T03:06:05.835421+00:00"}