Universality of the Local Spacing Distribution¶in Certain Ensembles of Hermitian Wigner Matrices

By K. Johansson. 2000

260 citations recorded by the source corpus.

Abstract: Consider an N×N hermitian random matrix with independent entries, not necessarily Gaussian, a so-called Wigner matrix. It has been conjectured that the local spacing distribution, i.e. the distribution of the distance between nearest neighbour eigenvalues in some part of the spectrum is, in the limit as N→∞, the same as that of hermitian random matrices from GUE. We prove this conjecture for a certain subclass of hermitian Wigner matrices.

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

View Paper: https://www.semanticscholar.org/paper/9d50087899fac8bf1edd03372f17ec0471edf243

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In Discrete Mathematics and Combinatorics

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Citations in the corpus

Regularity conditions in the CLT for linear eigenvalue statistics of Wigner matrices (2012, background): “As noted in [22], we have(2) gn(s, w) = 1 s f ′ n(w) + f ′ n(s)− f ′ n(w) s− w .”
Random Matrix Theory and Wireless Communications (2004, background): “This conjecture has been proved by [129] for a certain subclass of not necessarily Gaussian Wigner matrices.”
Wegner estimate and level repulsion for Wigner random matrices (2008, background): “Numerical evidences very strongly support these conjectures, nevertheless only a few rigorous results are known for ensembles without unitary invariance (notable exceptions are the universality of the Tracy-Widom distribution for the extremal eigenvalues [14] and the Wigner-Dyson sine-kernel for Wig”
Universality of random matrices and local relaxation flow (2009, methodology): “Johansson’s proof relied on the asymptotic analysis of an explicit formula by Brézin-Hikami [4, 15] for the correlation functions of the eigenvalues of Ĥ + sV .”
Random matrices: Universality of local eigenvalue statistics (2009, background): “The definition of Johansson matrices provides more degrees of freedom via the parameters t and M1n, and we can use this to remove the condition of the third and fourth moments.”
The largest eigenvalue of small rank perturbations of Hermitian random matrices (2004, methodology): “We refer the reader to (Johansson 2001) for a detailed proof of this.”
Eigenvector distribution of Wigner matrices (2011, methodology): “(iii) For the case of Hermitian Wigner matrices with a finite Gaussian component, it was proved in [19] that the correlation functions converge to those of GUE.”
Fixed Energy Universality for Generalized Wigner Matrices (2014, background): “5 Mesoscopic fluctuations for Gaussian ensembles This section follows Johansson’s method [23] to prove Gaussian fluctuations of linear statistics at any mesoscopic scale N−1+ε.”
The local relaxation flow approach to universality of the local statistics for random matrices (2009, methodology): “The result of [3,26] can be interpreted as stating that the local statistics of GUE is reached via DBM for time of order one.”
Universality of local eigenvalue statistics for some sample covariance matrices (2005, methodology): “Johansson’s computations show that the critical points forFu are given by the equation F ′ u(z) = (z − √ u)+ 2a2(z − √ z2 − 1) = 0, and the critical pointŝzc of Ĝu are just1i zc, wherezc are those ofFu.”
Deviation Inequalities on Largest Eigenvalues (2007, background): “[ Joha2 ] for an example of regularized Wigned matrices).”
Semicircle law on short scales and delocalization of eigenvectors for Wigner random matrices (2007, background): “This has been proven only for Gaussian and related models (see [7] and references therein) and for Wigner matrices where the distribution of the matrix elements were Gaussian convolutions [10].”
Bulk universality for Wigner matrices (2009, methodology): “We use the contour integral representation for the correlation functions of the eigenvalues of a matrix of the form H = Ĥ + aV , where V is a GUE matrix [3, 13].”
Random matrix theory in statistics: A review (2014, methodology): “First results of this kind were proved by Johansson (2001) and were subsequently improved by Ben Arous and Péché (2005) and Johansson (2009) in the setting of Gauss divisible Hermitian ensembles for which the matrices are in the form of a Wigner matrix perturbed by a Gaussian Wigner matrix.”
A Dynamical Approach to Random Matrix Theory (2017, methodology): “The first significant advance in this direction was made by Johansson [85], who proved the universality for complex Hermitian matrices under the assumption that the common distribution of the matrix entries has a substantial Gaussian component, i.e., the random matrix H is of the form H = H0 + aH G ”