Bulk universality for generalized Wigner matrices

Consider N × N Hermitian or symmetric random matrices H where the distribution of the ( i , j ) matrix element is given by a probability measure ν ij with a subexponential decay. Let be the variance for the probability measure ν ij with the normalization property that for all j . Under essentially t...

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Bibliographic Details
Published in:Probability theory and related fields 2012-10, Vol.154 (1-2), p.341-407
Main Authors: Erdős, László, Yau, Horng-Tzer, Yin, Jun
Format: Article
Language:English
Subjects:
Brownian motion
Decay
Economics
Eigenvalues
Finance
Gaussian
Insurance
Law
Management
Mathematical analysis
Mathematical and Computational Biology
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Matrices
Matrices (mathematics)
Matrix
Matrix methods
Operations Research/Decision Theory
Probability
Probability Theory and Stochastic Processes
Quantitative Finance
Random variables
Statistics
Statistics for Business
Studies
Theoretical
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Summary:Consider N × N Hermitian or symmetric random matrices H where the distribution of the ( i , j ) matrix element is given by a probability measure ν ij with a subexponential decay. Let be the variance for the probability measure ν ij with the normalization property that for all j . Under essentially the only condition that 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 .
ISSN:0178-8051
1432-2064
DOI:10.1007/s00440-011-0390-3