Speaker:
Albert Zhang, Courant Institute, New York University
Abstract:
The distribution of a single eigenvalue gap $\la_{k+1}(H_N) – \la_k(H_N)$ of a Wigner matrix is given asymptotically by the Gaudin-Mehta distribution in the bulk. While this convergence was known for matrices whose entries satisfy a four moment condition, the rates were non-explicit, given by $O(N^{-\eps})$ for some tiny $\eps > 0$. We show that the Green function comparison may be improved so that any Wigner matrix whose entries are supported on sufficiently many points will converge at rate $O(N^{-1/2 + \eps})$. Although this still does not yield quantitative universality for Bernoulli entries, previous works had only achieved reasonable rates for matrices with smooth entries.