# Bulk universality for generalized Wigner matrices

Erdös L, Yau H, Yin J. 2012. Bulk universality for generalized Wigner matrices. Probability Theory and Related Fields. 154(1–2), 341–407.

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Author

Erdös, László

^{ISTA}^{}; Yau, Horng-Tzer; Yin, JunAbstract

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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### Cite this

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/s00440-011-0390-3Erdös, L., Yau, H., & Yin, J. (2012). Bulk universality for generalized Wigner matrices.

*Probability Theory and Related Fields*. Springer. https://doi.org/10.1007/s00440-011-0390-3Erdös, László, Horng Yau, and Jun Yin. “Bulk Universality for Generalized Wigner Matrices.”

*Probability Theory and Related Fields*. Springer, 2012. 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. Springer, pp. 341–407, 2012.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.