Local single ring theorem on optimal scale

Let U and V be two independent N by N random matrices that are distributed according to Haar measure on U(N). Let Σ be a nonnegative deterministic N by N matrix. The single ring theorem [Ann. of Math. (2) 174 (2011) 1189-1217] asserts that the empirical eigenvalue distribution of the matrix X :=UΣV * converges weakly, in the limit of large N, to a deterministic measure which is supported on a single ring centered at the origin in . Within the bulk regime, that is, in the interior of the single ring, we establish the convergence of the empirical eigenvalue distribution on the optimal local scale of order N ^-1/2+ε and establish the optimal convergence rate. The same results hold true when U and V are Haar distributed on O(N).