{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:LXGXDDXFI4EZEJRS3REFOL36F3","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":"48901efbdac0773bbec6195ac2a3896131dbcd95da8d85c9fbd10e43bd645247","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2016-03-13T19:11:25Z","title_canon_sha256":"fbf5723e8a67c0bf130dabe89bffef59c719a6039193c30e61f473a4a89f185c"},"schema_version":"1.0","source":{"id":"1603.04066","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1603.04066","created_at":"2026-05-18T00:03:20Z"},{"alias_kind":"arxiv_version","alias_value":"1603.04066v2","created_at":"2026-05-18T00:03:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1603.04066","created_at":"2026-05-18T00:03:20Z"},{"alias_kind":"pith_short_12","alias_value":"LXGXDDXFI4EZ","created_at":"2026-05-18T12:30:29Z"},{"alias_kind":"pith_short_16","alias_value":"LXGXDDXFI4EZEJRS","created_at":"2026-05-18T12:30:29Z"},{"alias_kind":"pith_short_8","alias_value":"LXGXDDXF","created_at":"2026-05-18T12:30:29Z"}],"graph_snapshots":[{"event_id":"sha256:dd2620c41ffd59fc146e5cb137e50bbb3faad30ce76777bebcd2d2e26bd8fd1f","target":"graph","created_at":"2026-05-18T00:03:20Z","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":"It is well known that the spectral measure of eigenvalues of a rescaled square non-Hermitian random matrix with independent entries satisfies the circular law. We consider the product $TX$, where $T$ is a deterministic $N\\times M$ matrix and $X$ is a random $M\\times N$ matrix with independent entries having zero mean and variance $(N\\wedge M)^{-1}$. We prove a general local circular law for the empirical spectral distribution (ESD) of $TX$ at any point $z$ away from the unit circle under the assumptions that $N\\sim M$, and the matrix entries $X_{ij}$ have sufficiently high moments. More precis","authors_text":"Fan Yang, Haokai Xi, Jun Yin","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2016-03-13T19:11:25Z","title":"Local circular law for the product of a deterministic matrix with a random matrix"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.04066","kind":"arxiv","version":2},"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":"sha256:868a21ee57e44083ec2af6dc5b0a34c769472224f1fc3cc625c29773acd5d7c9","target":"record","created_at":"2026-05-18T00:03:20Z","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":"48901efbdac0773bbec6195ac2a3896131dbcd95da8d85c9fbd10e43bd645247","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2016-03-13T19:11:25Z","title_canon_sha256":"fbf5723e8a67c0bf130dabe89bffef59c719a6039193c30e61f473a4a89f185c"},"schema_version":"1.0","source":{"id":"1603.04066","kind":"arxiv","version":2}},"canonical_sha256":"5dcd718ee54709922632dc48572f7e2ed13d3374eb88ee655a88da89581f9dd5","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5dcd718ee54709922632dc48572f7e2ed13d3374eb88ee655a88da89581f9dd5","first_computed_at":"2026-05-18T00:03:20.049675Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:03:20.049675Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"GFnHQpt7SDd3fQZZr4LDU0nentojizox3PGe+Dn+B1zhzzewczuwL4VxTaECG8yo5UTDbg3IOhQmwvmBj/slBg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:03:20.050097Z","signed_message":"canonical_sha256_bytes"},"source_id":"1603.04066","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:868a21ee57e44083ec2af6dc5b0a34c769472224f1fc3cc625c29773acd5d7c9","sha256:dd2620c41ffd59fc146e5cb137e50bbb3faad30ce76777bebcd2d2e26bd8fd1f"],"state_sha256":"643298f7a2514a148044a9fd535ea55119ad64b2be40ccbba1c3f229888cc476"}