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We prove that the optimal (up to an extra $N^{{\\rm o}_N(1)}$ factor, where ${\\rm o}_N(1)$ can be arbitrarily small) eigenvalue rigidity holds. More precisely, denote $\\gamma_i$ as the classical location of the $i$-th eigenvalue under the Kesten-Mckay law in decreasing order. Then with probability $1-N^{-1+{\\rm o}_N(1)}$,\n  \\begin{align*}\n  |\\lambda_i-\\gamma_i|\\leq \\frac{N^{{\\rm"},"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":"2405.12161","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.PR","submitted_at":"2024-05-20T16:42:41Z","cross_cats_sorted":[],"title_canon_sha256":"4c8c7ef5176c2e2faa8a77a4e23fbd249a820060f69e8efb99eda94761df4c70","abstract_canon_sha256":"fe8219e12737b221dcdfdd951f6a99a8d5f812a7f176c384116fe8e247bd54a7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T08:21:00.910950Z","signature_b64":"R3fbhbP+HxH2t+d/I5E9O/kgd1kOP4ap/WP71HJc371683oSWV1eeN8PGADqHRXemTaOShfqIM64uvZDCGXBDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cb14900644c45fd4473dd32b23f95f97c27305cc5c3521cdd5a794ba74568dea","last_reissued_at":"2026-07-05T08:21:00.910474Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T08:21:00.910474Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Optimal Eigenvalue Rigidity of Random Regular Graphs","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Horng-Tzer Yau, Jiaoyang Huang, Theo McKenzie","submitted_at":"2024-05-20T16:42:41Z","abstract_excerpt":"Consider the normalized adjacency matrices of random $d$-regular graphs on $N$ vertices with fixed degree $d\\geq 3$, and denote the eigenvalues as $\\lambda_1=d/\\sqrt{d-1}\\geq \\lambda_2\\geq\\lambda_3\\cdots\\geq \\lambda_N$. We prove that the optimal (up to an extra $N^{{\\rm o}_N(1)}$ factor, where ${\\rm o}_N(1)$ can be arbitrarily small) eigenvalue rigidity holds. More precisely, denote $\\gamma_i$ as the classical location of the $i$-th eigenvalue under the Kesten-Mckay law in decreasing order. Then with probability $1-N^{-1+{\\rm o}_N(1)}$,\n  \\begin{align*}\n  |\\lambda_i-\\gamma_i|\\leq \\frac{N^{{\\rm"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2405.12161","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/2405.12161/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":"2405.12161","created_at":"2026-07-05T08:21:00.910540+00:00"},{"alias_kind":"arxiv_version","alias_value":"2405.12161v1","created_at":"2026-07-05T08:21:00.910540+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2405.12161","created_at":"2026-07-05T08:21:00.910540+00:00"},{"alias_kind":"pith_short_12","alias_value":"ZMKJABSEYRP5","created_at":"2026-07-05T08:21:00.910540+00:00"},{"alias_kind":"pith_short_16","alias_value":"ZMKJABSEYRP5IRZ5","created_at":"2026-07-05T08:21:00.910540+00:00"},{"alias_kind":"pith_short_8","alias_value":"ZMKJABSE","created_at":"2026-07-05T08:21:00.910540+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2412.20721","citing_title":"Ramanujan Graphs and Interlacing Families","ref_index":31,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/ZMKJABSEYRP5IRZ52MVSH6K7S7","json":"https://pith.science/pith/ZMKJABSEYRP5IRZ52MVSH6K7S7.json","graph_json":"https://pith.science/api/pith-number/ZMKJABSEYRP5IRZ52MVSH6K7S7/graph.json","events_json":"https://pith.science/api/pith-number/ZMKJABSEYRP5IRZ52MVSH6K7S7/events.json","paper":"https://pith.science/paper/ZMKJABSE"},"agent_actions":{"view_html":"https://pith.science/pith/ZMKJABSEYRP5IRZ52MVSH6K7S7","download_json":"https://pith.science/pith/ZMKJABSEYRP5IRZ52MVSH6K7S7.json","view_paper":"https://pith.science/paper/ZMKJABSE","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2405.12161&json=true","fetch_graph":"https://pith.science/api/pith-number/ZMKJABSEYRP5IRZ52MVSH6K7S7/graph.json","fetch_events":"https://pith.science/api/pith-number/ZMKJABSEYRP5IRZ52MVSH6K7S7/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ZMKJABSEYRP5IRZ52MVSH6K7S7/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ZMKJABSEYRP5IRZ52MVSH6K7S7/action/storage_attestation","attest_author":"https://pith.science/pith/ZMKJABSEYRP5IRZ52MVSH6K7S7/action/author_attestation","sign_citation":"https://pith.science/pith/ZMKJABSEYRP5IRZ52MVSH6K7S7/action/citation_signature","submit_replication":"https://pith.science/pith/ZMKJABSEYRP5IRZ52MVSH6K7S7/action/replication_record"}},"created_at":"2026-07-05T08:21:00.910540+00:00","updated_at":"2026-07-05T08:21:00.910540+00:00"}