Bulk universality for generalized Wigner matrices

L. Erdös, H. Yau, J. Yin, Probability Theory and Related Fields 154 (2012) 341–407.

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Abstract
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 σ ij 2 be the variance for the probability measure ν ij with the normalization property that Σ iσ i,j 2 = 1 for all j. Under essentially the only condition that c ≤ N σ ij 2 ≤ 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.
Publishing Year
Date Published
2012-10-01
Journal Title
Probability Theory and Related Fields
Volume
154
Issue
1-2
Page
341 - 407
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Erdös L, Yau H, Yin J. Bulk universality for generalized Wigner matrices. Probability Theory and Related Fields. 2012;154(1-2):341-407. doi:10.1007/s00440-011-0390-3
Erdös, L., Yau, H., & Yin, J. (2012). Bulk universality for generalized Wigner matrices. Probability Theory and Related Fields, 154(1–2), 341–407. https://doi.org/10.1007/s00440-011-0390-3
Erdös, László, Horng Yau, and Jun Yin. “Bulk Universality for Generalized Wigner Matrices.” Probability Theory and Related Fields 154, no. 1–2 (2012): 341–407. https://doi.org/10.1007/s00440-011-0390-3.
L. Erdös, H. Yau, and J. Yin, “Bulk universality for generalized Wigner matrices,” Probability Theory and Related Fields, vol. 154, no. 1–2, pp. 341–407, 2012.
Erdös L, Yau H, Yin J. 2012. Bulk universality for generalized Wigner matrices. Probability Theory and Related Fields. 154(1–2), 341–407.
Erdös, László, et al. “Bulk Universality for Generalized Wigner Matrices.” Probability Theory and Related Fields, vol. 154, no. 1–2, Springer, 2012, pp. 341–407, doi:10.1007/s00440-011-0390-3.

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