2010/07/27 by László Erdős, Laszlo Erdos, Horng‐Tzer Yau +5 · 46 citations
Mathematics · Physics and Astronomy · #15B52 #82B44 #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Combinatorics #Conjecture #Eigenvalues and eigenvectors #FOS: Mathematics #FOS: Physical sciences #Hermitian matrix #Mathematical Physics (math-ph) #Mathematical analysis #Mathematical physics #Mathematics #Order (exchange) #Physics #Probability (math.PR) #Pure mathematics #Quantum mechanics #Random Matrices and Applications #Random matrix #math-ph #math.MP #math.PR #msc:15B52 #msc:82B44
paper · pdf · doi:10.48550/arxiv.1007.4652
published in arXiv (Cornell University) (Cornell University) · 72 pages, no figures Sep 17,2011 a small error in the conditions of Lemma 5.1 was fixed and the argument in page 34-35 modified accordingly. On Oct 25 we added several explanation paragraphs and considerably expanded Section 7 to better illustrate the method
openalex publication_date 2010/07/27 · arxiv created 2011/10/26 · arxiv updated 2011/10/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Consider N× N hermitian or symmetric random matrices H with independent entries, where the distribution of the (i,j) matrix element is given by the probability measure νij with zero expectation and with variance σij2. We assume that the variances satisfy the normalization condition ∑i σ2ij = 1 for all j and that there is a positive constant c such that c≤ N σij2 ≤ c-1. We further assume that the probability distributions νij have a uniform subexponential decay. We prove that the Stieltjes transform of the empirical eigenvalue distribution of H is given by the Wigner semicircle law uniformly up to the edges of the spectrum with an error of order (N η)-1 where η is the imaginary part of the spectral parameter in the Stieltjes transform. There are three corollaries to this strong local semicircle law: (1) Rigidity of eigenvalues: If γj =γj,N denotes the \it classical location of the j-th eigenvalue under the semicircle law ordered in increasing order, then the j-th eigenvalue λj is close to γj in the sense that for any ξ>1 there is a constant L such that \mathbb P (∃ j : |λj-γj| ≥ (log N)L [ min ( j, N-j+1 ) ]-1/3 N-2/3 ) ≤ Cexp[-c(log N)ξ ] for N large enough. (2) The proof of the \it Dyson's conjecture \citeDy which states that the time scale of the Dyson Brownian motion to reach local equilibrium is of order N-1. (3) The edge universality holds in the sense that the probability distributions of the largest (and the smallest) eigenvalues of two generalized Wigner ensembles are the same in the large N limit provided that the second moments of the two ensembles are identical.