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For each $n\\geq2$ and $\\beta>0$, we obtain some inequalities on $\\mathbb{E}[p_{\\mu}(Z_n)\\bar{p_{\\nu}(Z_n)}]$, where $Z_n=(e^{i\\theta_1},\\ldots,e^{i\\theta_n})$ and $p_{\\mu}$ is the power-sum symmetric function for partition $\\mu$. When $\\beta=2$, our inequalities recover an identity by Diaconis and Evans for Haar-invariant unitary matrices. Further, we have the following: $ \\lim_{n\\t"},"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":"1102.4123","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-02-21T02:45:01Z","cross_cats_sorted":[],"title_canon_sha256":"ca482ff3c934433d1cff08d499035079dd161bcfd3055224a31731d48d0e908e","abstract_canon_sha256":"c176dc732503c3901dcb2339f3861da23ea0c825760ac67f55c76df69825be36"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:23:59.287470Z","signature_b64":"cuhJmgMH5iIzpU6YBSymq85ZuZnNdjxCY4+fIP40HyI5m6bW+DeYduP3cJHN8NyMegCT/0+KrooSe42g8MlhBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2b1684f3fa969f1070e3ae8e8f51efdf28cb5df625b624774771dd843b410b42","last_reissued_at":"2026-05-18T01:23:59.286799Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:23:59.286799Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Moments of traces of circular beta-ensembles","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Sho Matsumoto, Tiefeng Jiang","submitted_at":"2011-02-21T02:45:01Z","abstract_excerpt":"Let $\\theta_1,\\ldots,\\theta_n$ be random variables from Dyson's circular $\\beta$-ensemble with probability density function $\\operatorname {Const}\\cdot\\prod_{1\\leq j<k\\leq n}|e^{i\\theta_j}-e^{i\\theta _k}|^{\\beta}$. For each $n\\geq2$ and $\\beta>0$, we obtain some inequalities on $\\mathbb{E}[p_{\\mu}(Z_n)\\bar{p_{\\nu}(Z_n)}]$, where $Z_n=(e^{i\\theta_1},\\ldots,e^{i\\theta_n})$ and $p_{\\mu}$ is the power-sum symmetric function for partition $\\mu$. When $\\beta=2$, our inequalities recover an identity by Diaconis and Evans for Haar-invariant unitary matrices. Further, we have the following: $ \\lim_{n\\t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1102.4123","kind":"arxiv","version":2},"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":"1102.4123","created_at":"2026-05-18T01:23:59.286902+00:00"},{"alias_kind":"arxiv_version","alias_value":"1102.4123v2","created_at":"2026-05-18T01:23:59.286902+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1102.4123","created_at":"2026-05-18T01:23:59.286902+00:00"},{"alias_kind":"pith_short_12","alias_value":"FMLIJ472S2PR","created_at":"2026-05-18T12:26:28.662955+00:00"},{"alias_kind":"pith_short_16","alias_value":"FMLIJ472S2PRA4HD","created_at":"2026-05-18T12:26:28.662955+00:00"},{"alias_kind":"pith_short_8","alias_value":"FMLIJ472","created_at":"2026-05-18T12:26:28.662955+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/FMLIJ472S2PRA4HDV2HI6UPP34","json":"https://pith.science/pith/FMLIJ472S2PRA4HDV2HI6UPP34.json","graph_json":"https://pith.science/api/pith-number/FMLIJ472S2PRA4HDV2HI6UPP34/graph.json","events_json":"https://pith.science/api/pith-number/FMLIJ472S2PRA4HDV2HI6UPP34/events.json","paper":"https://pith.science/paper/FMLIJ472"},"agent_actions":{"view_html":"https://pith.science/pith/FMLIJ472S2PRA4HDV2HI6UPP34","download_json":"https://pith.science/pith/FMLIJ472S2PRA4HDV2HI6UPP34.json","view_paper":"https://pith.science/paper/FMLIJ472","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1102.4123&json=true","fetch_graph":"https://pith.science/api/pith-number/FMLIJ472S2PRA4HDV2HI6UPP34/graph.json","fetch_events":"https://pith.science/api/pith-number/FMLIJ472S2PRA4HDV2HI6UPP34/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/FMLIJ472S2PRA4HDV2HI6UPP34/action/timestamp_anchor","attest_storage":"https://pith.science/pith/FMLIJ472S2PRA4HDV2HI6UPP34/action/storage_attestation","attest_author":"https://pith.science/pith/FMLIJ472S2PRA4HDV2HI6UPP34/action/author_attestation","sign_citation":"https://pith.science/pith/FMLIJ472S2PRA4HDV2HI6UPP34/action/citation_signature","submit_replication":"https://pith.science/pith/FMLIJ472S2PRA4HDV2HI6UPP34/action/replication_record"}},"created_at":"2026-05-18T01:23:59.286902+00:00","updated_at":"2026-05-18T01:23:59.286902+00:00"}