Bulk universality for generalized Wigner matrices

By L. Erdős, Horng-Tzer Yau, Jun Yin. 2010

324 citations recorded by the source corpus.

Consider N × N Hermitian or symmetric random matrices H where the distribution of the (i, j) matrix element is given by a probability measure νij with a subexponential decay. Let $${\sigma_{ij}^2}$$ be the variance for the probability measure νij with the normalization property that $${\sum_{i} \sigma^2_{ij} = 1}$$ for all j. Under essentially the only condition that $${c\le N \sigma_{ij}^2 \le c^{-1}}$$ for some constant c > 0, we prove that, in the limit N → ∞, the eigenvalue spacing statistics of H in the bulk of the spectrum coincide with those of the Gaussian unitary or orthogonal ensemble (GUE or GOE). We also show that for band matrices with bandwidth M the local semicircle law holds to the energy scale M−1.

Subfields: Discrete Mathematics and Combinatorics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/8356ef91577488c5b2bd142f2c2944231aa1d060

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In Discrete Mathematics and Combinatorics

In Mathematical Physics

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Citations in the corpus

Random matrices: Universal properties of eigenvectors (2011, background): “ M n −1 pq ! −EG 1 √ n M′ −1 pq ! = O(n−c0). We prove Theorem 9 in Section 5; it is established by combining Theorem 8 with an eigenvalue rigidity result from [10] and a local semicircle law from [9]. This result generalizes a similar four moment theorem from ([9] ( [9, Theorem 2.3]). Its main stre”
On the principal components of sample covariance matrices (2014, methodology): “We shall use the Green function comparison method [22, 23, 29] to prove Lemma 7.”
The local relaxation flow approach to universality of the local statistics for random matrices (2009, methodology): “dividual eigenvalues for two diﬀerent ensembles. For our application to the correlation functions and gap distributions, an alternative approach is to use the recent Green function comparison theorem [21]. This will also remove the smoothness and logarithmic Sobolev inequality restrictions in Theore”
Universality of random matrices and local relaxation flow (2009, background): “te dimensional, there are enough freedom to choose perturbations so as to match the ﬁrst four moments as long as m4 >m2 3 + 1. An elementary detailed proof of this fact is 8 given in Lemma C.1. of [15]. Therefore, bνsatisﬁes the assumption of Theorem 2.3 and thus (2.19) holds for the measure νb. ”
The Isotropic Semicircle Law and Deformation of Wigner Matrices (2011, unknown): “To avoid confusion, we remark that the Stieltjes transformm was denoted bymsc in the papers [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], in which m had a different meaning from (2.”
The local semicircle law for a general class of random matrices (2012, background): “ Nrandom matrices whose entries h ijare independent up to a symmetry constraint, but not necessarily identically distributed. Our main result is a local semicircle law which improves previous results [17] both in the bulk and at the edge. The error bounds are given in terms of the basic small parame”
Universality of local spectral statistics of random matrices (2011, methodology): “l with the generalized Wigner matrices, we needed to consider the self-consistent equation of Gij(z), the matrix elements of the Green function, since there is no closed equation for m(z) = N−1TrG(z) [36, 37]. In particular, this method implied the optimal rigidity estimate of eigenvalues in the bul”
Regularity conditions in the CLT for linear eigenvalue statistics of Wigner matrices (2012, background): “The last quantity is bounded by e−n c ([16], Lemma 7.”
Eigenvector distribution of Wigner matrices (2011, background): “We prove that the joint probability distribution of the components of eigenvectors associated with eigenvalues close to the spectral edge agrees with that of the corresponding Gaussian ensemble.”
Delocalization of eigenvectors of random matrices with independent entries (2013, background): “(i) We can represent the distance as d(DX,Ek) = ‖BX‖2 where B = PE⊥k D.”
Tracy-Widom law for the extreme eigenvalues of sample correlation matrices (2011, unknown): “Since the proof of (4.14) is similar to the counterpart in [14], we will only state the proof for k = 3 below.”
A Dynamical Approach to Random Matrix Theory (2017, methodology): “The following lemma in a slightly less general forms appeared in [59, 68, 69].”
Tracy-Widom Distribution for the Largest Eigenvalue of Real Sample Covariance Matrices with General Population (2014, methodology): “Optimal local laws in the bulk and at the edges of the spectrum were obtained for Wigner matrices in [18, 17, 15].”