On fluctuations of global and mesoscopic linear statistics of generalized Wigner matrices

Abstract We prove that the energy of any eigenvector of a sum of several independent large Wigner matrices is equally distributed among these matrices with very high precision. This shows a particularly strong microcanonical form of the equipartition principle for quantum systems whose components are modelled by Wigner matrices.

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Wigner Matrices and Semicircular Law

Springer Series in Statistics - Spectral Analysis of Large Dimensional Random Matrices ◽

10.1007/978-1-4419-0661-8_2 ◽

2009 ◽

pp. 15-38 ◽

Cited By ~ 2

Author(s):

Zhidong Bai ◽

Jack W. Silverstein

Keyword(s):

Wigner Matrices ◽

Semicircular Law

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A comment on the Wigner-Dyson-Mehta bulk universality conjecture for Wigner matrices

Electronic Journal of Probability ◽

10.1214/ejp.v17-1779 ◽

2012 ◽

Vol 17 (0) ◽

Cited By ~ 5

Author(s):

László Erdős ◽

Horng-Tzer Yau

Keyword(s):

Wigner Matrices

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GLOBAL FLUCTUATIONS FOR LINEAR STATISTICS OF β-JACOBI ENSEMBLES

Random Matrices Theory and Application ◽

10.1142/s201032631250013x ◽

2012 ◽

Vol 01 (04) ◽

pp. 1250013 ◽

Cited By ~ 11

Author(s):

IOANA DUMITRIU ◽

ELLIOT PAQUETTE

Keyword(s):

Gaussian Process ◽

Covariance Matrix ◽

Law Of Large Numbers ◽

Test Functions ◽

Linear Statistics ◽

Jacobi Ensemble ◽

Large Numbers ◽

The Mean ◽

Shifted Chebyshev Polynomial ◽

Global Fluctuations

We study the global fluctuations for linear statistics of the form [Formula: see text] as n → ∞, for C1 functions f, and λ1, …, λn being the eigenvalues of a (general) β-Jacobi ensemble. The fluctuation from the mean [Formula: see text] turns out to be given asymptotically by a Gaussian process. We compute the covariance matrix for the process and show that it is diagonalized by a shifted Chebyshev polynomial basis; in addition, we analyze the deviation from the predicted mean for polynomial test functions, and we obtain a law of large numbers.

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