{"paper":{"title":"Two CLTs for Sparse Random Matrices","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Simona Diaconu","submitted_at":"2022-10-18T06:41:59Z","abstract_excerpt":"Let $G=G(n,p_n)$ be a homogeneous Erd\\\"os-R\\'enyi graph, and $A$ its adjacency matrix with eigenvalues $\\lambda_1(A) \\geq \\lambda_2(A) \\geq ... \\geq \\lambda_n(A).$ Local laws have been used to show that $lambda_2(A)$ can exhibit fundamentally different behaviors: Tracy-Widom ($p_n \\gg n^{-2/3}$), normal ($n^{-7/9} \\ll p_n \\ll~n^{-2/3}$), and a mix of both ($p_n=cn^{-2/3}$). Additionally, this technique renders the largest eigenvalue $\\lambda_1(A),$ separated from the rest of the spectrum for $p_n \\gg n^{-1},$ has Gaussian fluctuations when $p_n \\geq n^{-1}(\\log{n})^{6+c}$ for some $c>0.$ This "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2210.09625","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2210.09625/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"}