{"paper":{"title":"Local Marchenko-Pastur Law at the Hard Edge of Sample Covariance Matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MP","math.PR"],"primary_cat":"math-ph","authors_text":"Anna Maltsev, Benjamin Schlein, Claudio Cacciapuoti","submitted_at":"2012-06-08T11:21:54Z","abstract_excerpt":"Let $X_N$ be a $N\\times N$ matrix whose entries are i.i.d. complex random variables with mean zero and variance $\\frac{1}{N}$. We study the asymptotic spectral distribution of the eigenvalues of the covariance matrix $X_N^*X_N$ for $N\\to\\infty$. We prove that the empirical density of eigenvalues in an interval $[E,E+\\eta]$ converges to the Marchenko-Pastur law locally on the optimal scale, $N \\eta /\\sqrt{E} \\gg (\\log N)^b$, and in any interval up to the hard edge, $\\frac{(\\log N)^b}{N^2}\\lesssim E \\leq 4-\\kappa$, for any $\\kappa >0$. As a consequence, we show the complete delocalization of the"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1206.1730","kind":"arxiv","version":3},"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"}