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We prove that for any $k$, if $f$ is smooth enough, there exist deterministic constants $\\alpha_i^P(f,Z^N)$ such that $$ \\mathbb{E}\\left[\\frac{1}{N}\\text{Tr}\\left( f(P(X^N,Z^N)) \\right)\\right]\\ =\\ \\sum_{i=0}^k \\frac{\\alpha_i^P(f,Z^N)}{N^{2i}}\\ +\\ \\mathcal{O}(N^{-2k-2}) .$$ Besides the constants $\\alpha_i^P(f,Z^N)$ are built explicitly with the help of free probability."},"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":"2011.04146","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2020-11-09T02:12:03Z","cross_cats_sorted":["math.OA"],"title_canon_sha256":"2a04911327f499dc333d69f4abee9d4129e66afc486a8ce58be53c41eccde04e","abstract_canon_sha256":"5b6f960ad5f828868292af1289aa854857b377de2a64977f6ea5f9d72a9ea27c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T05:23:06.051493Z","signature_b64":"T5pFhDITnU5NuteEYAtnWctZ1/UWnuv/p+g6y/1UZg/v4rk+wV2ltHlL1XU23VCfOG43WHyfS2c7ZSxMvRyfDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8ab00e7d3ead66d1c0bda0f9dcfdd45e3a9ece4f392264630d7f03226f207879","last_reissued_at":"2026-07-05T05:23:06.051016Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T05:23:06.051016Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Asymptotic expansion of smooth functions in polynomials in deterministic matrices and iid GUE matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.OA"],"primary_cat":"math.PR","authors_text":"Felix Parraud","submitted_at":"2020-11-09T02:12:03Z","abstract_excerpt":"Let $X^N$ be a family of $N\\times N$ independent GUE random matrices, $Z^N$ a family of deterministic matrices, $P$ a self-adjoint non-commutative polynomial, that is for any $N$, $P(X^N)$ is self-adjoint, $f$ a smooth function. We prove that for any $k$, if $f$ is smooth enough, there exist deterministic constants $\\alpha_i^P(f,Z^N)$ such that $$ \\mathbb{E}\\left[\\frac{1}{N}\\text{Tr}\\left( f(P(X^N,Z^N)) \\right)\\right]\\ =\\ \\sum_{i=0}^k \\frac{\\alpha_i^P(f,Z^N)}{N^{2i}}\\ +\\ \\mathcal{O}(N^{-2k-2}) .$$ Besides the constants $\\alpha_i^P(f,Z^N)$ are built explicitly with the help of free probability."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2011.04146","kind":"arxiv","version":4},"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/2011.04146/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":"2011.04146","created_at":"2026-07-05T05:23:06.051069+00:00"},{"alias_kind":"arxiv_version","alias_value":"2011.04146v4","created_at":"2026-07-05T05:23:06.051069+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2011.04146","created_at":"2026-07-05T05:23:06.051069+00:00"},{"alias_kind":"pith_short_12","alias_value":"RKYA47J6VVTN","created_at":"2026-07-05T05:23:06.051069+00:00"},{"alias_kind":"pith_short_16","alias_value":"RKYA47J6VVTNDQF5","created_at":"2026-07-05T05:23:06.051069+00:00"},{"alias_kind":"pith_short_8","alias_value":"RKYA47J6","created_at":"2026-07-05T05:23:06.051069+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2412.09290","citing_title":"Asymptotics of Harish-Chandra transform and infinitesimal freeness","ref_index":38,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/RKYA47J6VVTNDQF5UD45Z7OULY","json":"https://pith.science/pith/RKYA47J6VVTNDQF5UD45Z7OULY.json","graph_json":"https://pith.science/api/pith-number/RKYA47J6VVTNDQF5UD45Z7OULY/graph.json","events_json":"https://pith.science/api/pith-number/RKYA47J6VVTNDQF5UD45Z7OULY/events.json","paper":"https://pith.science/paper/RKYA47J6"},"agent_actions":{"view_html":"https://pith.science/pith/RKYA47J6VVTNDQF5UD45Z7OULY","download_json":"https://pith.science/pith/RKYA47J6VVTNDQF5UD45Z7OULY.json","view_paper":"https://pith.science/paper/RKYA47J6","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2011.04146&json=true","fetch_graph":"https://pith.science/api/pith-number/RKYA47J6VVTNDQF5UD45Z7OULY/graph.json","fetch_events":"https://pith.science/api/pith-number/RKYA47J6VVTNDQF5UD45Z7OULY/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/RKYA47J6VVTNDQF5UD45Z7OULY/action/timestamp_anchor","attest_storage":"https://pith.science/pith/RKYA47J6VVTNDQF5UD45Z7OULY/action/storage_attestation","attest_author":"https://pith.science/pith/RKYA47J6VVTNDQF5UD45Z7OULY/action/author_attestation","sign_citation":"https://pith.science/pith/RKYA47J6VVTNDQF5UD45Z7OULY/action/citation_signature","submit_replication":"https://pith.science/pith/RKYA47J6VVTNDQF5UD45Z7OULY/action/replication_record"}},"created_at":"2026-07-05T05:23:06.051069+00:00","updated_at":"2026-07-05T05:23:06.051069+00:00"}