{"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"}