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We prove that the normalized row-ordered permanent $W_n^{\\mathbb{K}}=\\operatorname{per}_{\\mathbb{K}}G_n^{\\mathbb{K}}/\\sqrt{n!}$ has a radial density $p_n^{\\mathbb{K}}$ satisfying $\\|p_n^{\\mathbb{K}}\\|_\\infty=p_n^{\\mathbb{K}}(0)\\lesssim_\\beta n^{(\\beta+2)/4}$ and $\\sup_{z\\in\\mathbb{K}}\\mathbb{P}(|W_n^{\\mathbb{K}}-z|\\leq\\varepsilon)\\lesssim_\\beta n^{("},"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":"2607.20329","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2026-07-22T16:15:28Z","cross_cats_sorted":["cs.CC","math-ph","math.MP","quant-ph"],"title_canon_sha256":"dec1e0d7d81a56c20555bebe67c498d5a22d86ab3c7f6e5c1fc6de6eff556acb","abstract_canon_sha256":"066eaed639efaf6ed6dcdfed70db9542cea2175580c1e0d991d7a1f048615620"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-23T01:25:14.687495Z","signature_b64":"hp2SnWoD/V0oluXTham0HID0MSSt6RF2N1DwFmna3ltoXUzPlx53dUNZr1Jum4pyPN3xZdxYaBpqw7TpvyqfAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"abd96c5ca162369aae83a2da59168d7ff543a676009a4ff2b498ce289550542d","last_reissued_at":"2026-07-23T01:25:14.686615Z","signature_status":"signed_v1","first_computed_at":"2026-07-23T01:25:14.686615Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Anticoncentration of the Permanent in Ginibre Ensembles","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CC","math-ph","math.MP","quant-ph"],"primary_cat":"math.PR","authors_text":"Frederic Koehler, Pui Kuen Leung","submitted_at":"2026-07-22T16:15:28Z","abstract_excerpt":"Let $\\mathbb{K}\\in\\{\\mathbb{R},\\mathbb{C},\\mathbb{H}\\}$, put $\\beta=\\dim_{\\mathbb{R}}\\mathbb{K}$, and let $G_n^{\\mathbb{K}}$ be an $n\\times n$ matrix with i.i.d. standard $\\mathbb{K}$-Gaussian entries, namely a standard $\\mathbb{K}$-Ginibre matrix. We prove that the normalized row-ordered permanent $W_n^{\\mathbb{K}}=\\operatorname{per}_{\\mathbb{K}}G_n^{\\mathbb{K}}/\\sqrt{n!}$ has a radial density $p_n^{\\mathbb{K}}$ satisfying $\\|p_n^{\\mathbb{K}}\\|_\\infty=p_n^{\\mathbb{K}}(0)\\lesssim_\\beta n^{(\\beta+2)/4}$ and $\\sup_{z\\in\\mathbb{K}}\\mathbb{P}(|W_n^{\\mathbb{K}}-z|\\leq\\varepsilon)\\lesssim_\\beta n^{("},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.20329","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.20329/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2607.20329","created_at":"2026-07-23T01:25:14.687081+00:00"},{"alias_kind":"arxiv_version","alias_value":"2607.20329v1","created_at":"2026-07-23T01:25:14.687081+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.20329","created_at":"2026-07-23T01:25:14.687081+00:00"},{"alias_kind":"pith_short_12","alias_value":"VPMWYXFBMI3J","created_at":"2026-07-23T01:25:14.687081+00:00"},{"alias_kind":"pith_short_16","alias_value":"VPMWYXFBMI3JVLUD","created_at":"2026-07-23T01:25:14.687081+00:00"},{"alias_kind":"pith_short_8","alias_value":"VPMWYXFB","created_at":"2026-07-23T01:25:14.687081+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/VPMWYXFBMI3JVLUDULNFSFUNP7","json":"https://pith.science/pith/VPMWYXFBMI3JVLUDULNFSFUNP7.json","graph_json":"https://pith.science/api/pith-number/VPMWYXFBMI3JVLUDULNFSFUNP7/graph.json","events_json":"https://pith.science/api/pith-number/VPMWYXFBMI3JVLUDULNFSFUNP7/events.json","paper":"https://pith.science/paper/VPMWYXFB"},"agent_actions":{"view_html":"https://pith.science/pith/VPMWYXFBMI3JVLUDULNFSFUNP7","download_json":"https://pith.science/pith/VPMWYXFBMI3JVLUDULNFSFUNP7.json","view_paper":"https://pith.science/paper/VPMWYXFB","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2607.20329&json=true","fetch_graph":"https://pith.science/api/pith-number/VPMWYXFBMI3JVLUDULNFSFUNP7/graph.json","fetch_events":"https://pith.science/api/pith-number/VPMWYXFBMI3JVLUDULNFSFUNP7/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/VPMWYXFBMI3JVLUDULNFSFUNP7/action/timestamp_anchor","attest_storage":"https://pith.science/pith/VPMWYXFBMI3JVLUDULNFSFUNP7/action/storage_attestation","attest_author":"https://pith.science/pith/VPMWYXFBMI3JVLUDULNFSFUNP7/action/author_attestation","sign_citation":"https://pith.science/pith/VPMWYXFBMI3JVLUDULNFSFUNP7/action/citation_signature","submit_replication":"https://pith.science/pith/VPMWYXFBMI3JVLUDULNFSFUNP7/action/replication_record"}},"created_at":"2026-07-23T01:25:14.687081+00:00","updated_at":"2026-07-23T01:25:14.687081+00:00"}