Bulk universality for Wigner matrices

L. Erdös, J. Ramírez, H. Yau, S. Péché, B. Schlein, Communications on Pure and Applied Mathematics 63 (2010) 895–925.

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Abstract
We consider N ×N Hermitian Wigner random matrices H where the probabilitydensity for each matrix element is given by the density v(x)=e-U(x). We prove that the eigenvalue statistics in the bulk are given by the Dyson sine kernel provided that U ∈ C 6(R{double-struck}) with at most polynomially growing derivatives and v(x)≤C e-c(x) for x large. The proof is based upon an approximate time reversal of the Dyson Brownian motion combined with the convergence of the eigenvalue density to the Wigner semicircle law on short scales.
Publishing Year
Date Published
2010-07-01
Journal Title
Communications on Pure and Applied Mathematics
Volume
63
Issue
7
Page
895 - 925
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Erdös L, Ramírez J, Yau H, Péché S, Schlein B. Bulk universality for Wigner matrices. Communications on Pure and Applied Mathematics. 2010;63(7):895-925. doi:10.1002/cpa.20317
Erdös, L., Ramírez, J., Yau, H., Péché, S., & Schlein, B. (2010). Bulk universality for Wigner matrices. Communications on Pure and Applied Mathematics, 63(7), 895–925. https://doi.org/10.1002/cpa.20317
Erdös, László, José Ramírez, Horng Yau, Sandrine Péché, and Benjamin Schlein. “Bulk Universality for Wigner Matrices.” Communications on Pure and Applied Mathematics 63, no. 7 (2010): 895–925. https://doi.org/10.1002/cpa.20317.
L. Erdös, J. Ramírez, H. Yau, S. Péché, and B. Schlein, “Bulk universality for Wigner matrices,” Communications on Pure and Applied Mathematics, vol. 63, no. 7, pp. 895–925, 2010.
Erdös L, Ramírez J, Yau H, Péché S, Schlein B. 2010. Bulk universality for Wigner matrices. Communications on Pure and Applied Mathematics. 63(7), 895–925.
Erdös, László, et al. “Bulk Universality for Wigner Matrices.” Communications on Pure and Applied Mathematics, vol. 63, no. 7, Wiley-Blackwell, 2010, pp. 895–925, doi:10.1002/cpa.20317.

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