1308.1057 (Sean O'Rourke et al.)

Universality of local eigenvalue statistics in random matrices with
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Sean O'Rourke, Van Vu

Consider a random matrix of the form W_n = M_n + D_n, where M_n is a Wigner matrix and D_n is a real deterministic diagonal matrix (D_n is commonly referred to as an external source in the mathematical physics literature). We study the universality of the local eigenvalue statistics of W_n for a general class of Wigner matrices M_n and diagonal matrices D_n. Unlike the setting of many recent results concerning universality, the global semicircle law fails for this model. However, we can still obtain the universal sine kernel formula for the correlation functions. This demonstrates the remarkable phenomenon that local laws are more resilient than global ones. The universality of the correlation functions follows from a four moment theorem, which we prove using a variant of the approach used earlier by Tao and Vu.

View original: http://arxiv.org/abs/1308.1057