Rigidity of eigenvalues of generalized Wigner matrices
By L. Erdős, Horng-Tzer Yau, Jun Yin. 2010
445 citations recorded by the source corpus.
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 ∑iσij2=1 for all j and that there is a positive constant c such that c⩽Nσij2⩽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 (Nη)−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 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 some positive constants C, c P(∃j:|λj−γj|⩾(logN)CloglogN[min(j,N−j+1)]−1/3N−2/3)⩽Cexp[−(logN)cloglogN] for N large enough. (2) The proof of Dysonʼs conjecture (Dyson, 1962 [15]) which states that the time scale of the Dyson Brownian motion to reach local equilibrium is of order N−1 up to logarithmic corrections. (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.
Subfields: Discrete Mathematics and Combinatorics, Mathematical Physics, Statistics and Probability
View Paper: https://www.semanticscholar.org/paper/9ec80717bba00962d1a9dd73904036e5aca94945
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Citations in the corpus
- Universality for the largest eigenvalue of sample covariance matrices with general population (2013, methodology): “The main route to prove the universality properties in [18, 19, 31] can be summarized into the following two steps.”
- A necessary and sufficient condition for edge universality of Wigner matrices (2012, methodology): “In the section 6 of [26], the edge universality problem was converted into proving Theorem 6.”
- The local semicircle law for a general class of random matrices (2012, methodology): “These estimates underlie the derivation of rigidity estimates for individual eigenvalues, which in turn were used in [16] to prove Dyson’s conjecture on the optimal local relaxation time for the Dyson Brownian motion.”
- TAIL BOUNDS FOR ALL EIGENVALUES OF A SUM OF RANDOM MATRICES (2011, background): “Under the assumption that the entries exhibit uniform subgaussian decay, Erdös, Yau, and Yin have strengthened this result by showing that, up to log factors, the eigenvalues of nAn are within O(n −2/3) of their classical position with high probability [EYY10].”
- Anisotropic local laws for random matrices (2014, background): “[20] Erdős, L; Yau, H-T; Yin, J, Rigidity of eigenvalues of generalized Wigner matrices, Adv.”
- The Isotropic Semicircle Law and Deformation of Wigner Matrices (2011, methodology): “The strongest versions of the local semicircle law, proved in [15, 16, 27], give precise estimates on the local eigenvalue density, down to scales containing N " eigenvalues.”
- Universality of covariance matrices (2011, methodology): “For proving bulk universality of eigenvalues, we follow the general approach for the universality of Gaussian divisible ensembles [17, 18, 10, 11, 25, 26] by embedding the covariance matrix into a stochastic flow of matrices and so that the eigenvalues evolve according to a distinguished coupled sys”
- Lectures on the local semicircle law for Wigner matrices (2016, unknown): “04 05 5v 3 [ m at h. PR ] 1 8 O ct 2 01 6 Contents”
- Regularity conditions in the CLT for linear eigenvalue statistics of Wigner matrices (2012, methodology): “41) in [14], we have ∑ i 6=j |G ij | 2 1/2 ≤ (∑”
- A Dynamical Approach to Random Matrix Theory (2017, methodology): “A different proof (with a better bound on the constants) was given in [70].”
- Edge universality for deformed Wigner matrices (2014, methodology): “2 in [19], are already proved in Lemma 4.”
- Random matrices: Sharp concentration of eigenvalues (2012, background): “s sc(z)+o(1) and (45) R(z)ii = s sc(z) +o(1) outside of an event with probability O(nO(1)exp(−c(nη)c)). We note that this corollary is essentially [17, Theorem 3.1]; in the statement of the result in [17] the additional constraint η ≥ logC loglogn /n for some constant C is imposed, but this constrai”