Universality of Wigner random matrices: a survey of recent results

By L. Erdős. 2010

190 citations recorded by the source corpus.

This is a study of the universality of spectral statistics for large random matrices. Considered are symmetric, Hermitian, or quaternion self-dual random matrices with independent identically distributed entries (Wigner matrices), where the probability distribution of each matrix element is given by a measure with zero expectation and with subexponential decay. The main result is that the correlation functions of the local eigenvalue statistics in the bulk of the spectrum coincide with those of the Gaussian Orthogonal Ensemble (GOE), the Gaussian Unitary Ensemble (GUE), and the Gaussian Symplectic Ensemble (GSE), respectively, in the limit as . This approach is based on a study of the Dyson Brownian motion via a related new dynamics, the local relaxation flow. As a main input, it is established that the density of the eigenvalues converges to the Wigner semicircle law, and this holds even down to the smallest possible scale. Moreover, it is shown that the eigenvectors are completely delocalized. These results hold even without the condition that the matrix elements are identically distributed: only independence is used. In fact, for the matrix elements of the Green function strong estimates are given that imply that the local statistics of any two ensembles in the bulk are identical if the first four moments of the matrix elements match. Universality at the spectral edges requires matching only two moments. A Wigner-type estimate is also proved, and it is shown that the eigenvalues repel each other on arbitrarily small scales. Bibliography: 108 titles.

Subfields: Artificial Intelligence, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/e7f69e0248241b4771193118c4fa4fd8224a8b1b

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Citations in the corpus

Localization and delocalization of eigenvectors for heavy-tailed random matrices (2012, background): “[3–5,8,15,28] for reviews, or [20,21,23,27,29] for recent results on universality.”
Universality for Random Tensors (2011, background): “Although universality can be established under quite general assumptions, in the matrix case there exist invariant probability laws which are not universal [13].”
Universality for the largest eigenvalue of sample covariance matrices with general population (2013, background): “Such a limiting law on microscopic scales was developed in a series of work [12, 13, 14, 19] for Wigner matrices originally and was shown to be crucial in recent works on universality problems of local eigenvalue statistics, one can refer to [15, 18, 34] and the survey paper [10] for instance.”
Random matrices: Universality of local spectral statistics of non-Hermitian matrices (2012, methodology): “ing conditions have now largely been removed, the exponential decay condition relaxed to a ﬁnite moment condition, and the bulk results extended to the edge; see the discussion in [53] or the surveys [14], [26], [40], [56] for surveys for more details. In view of these results, it is reasonable to c”
Regularity conditions in the CLT for linear eigenvalue statistics of Wigner matrices (2012, background): “Yau and their collaborators have obtained large deviation bounds of order η−2n2 log n for the trace of the resolvent TrG(z) for η n−1 (see the survey paper [11]).”
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”
Inflation and String Theory (2014, background): “[461,465] — but to simplify the discussion we will continue to omit the associated technicalities.”
Polynomial convolutions and (nite) free probability (2021, methodology): “Polynomials have also been used in similar ways in a strictly free probabilistic setting [2], and are one of the major tools in the investigation of universality for random matrices [12].”