Bulk universality for generalized Wigner matrices

Erdős, László; Yau, Horng-Tzer; Yin, Jun
October 2012
Probability Theory & Related Fields;Oct2012, Vol. 154 Issue 1/2, p341
Academic Journal
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 with a subexponential decay. Let $${\sigma_{ij}^2}$$ be the variance for the probability measure ν with the normalization property that $${\sum_{i} \sigma^2_{ij} = 1}$$ for all j. Under essentially the only condition that $${c\le N \sigma_{ij}^2 \le c^{-1}}$$ 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.


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