{"paper":{"title":"Quantitative Tracy-Widom laws for sparse random matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Kevin Schnelli, Teodor Bucht, Yuanyuan Xu","submitted_at":"2025-07-25T14:50:07Z","abstract_excerpt":"We consider the fluctuations of the largest eigenvalue of sparse random matrices, the class of random matrices that includes the normalized adjacency matrices of the Erd\\H{o}s-R\\'enyi graph $G(N, p)$. We show that the fluctuations of the largest eigenvalue converge to the Tracy-Widom law at a rate almost $O(N^{-1/3 } + p^{-2} N^{-4/3})$ in the regime $p \\gg N^{-2/3 }$. Our proof builds upon the Green function comparison method initiated by Erd\\H{o}s, Yau, and Yin [22]. To show a Green function comparison theorem for fine spectral scales, we implement algorithms for symbolic computations involv"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.19340","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/2507.19340/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"}