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 $\Sigma$ be a non-negative deterministic $N$ by $N$ matrix. The single ring theorem [26] asserts that the empirical eigenvalue distribution of the matrix $X:= U\Sigma V^*$ converges weakly, in the limit of large $N$, to a deterministic measure which is supported on a single ring centered at the origin in $\mathbb{C}$. Within the bulk regime, i.e. in the interior of the single ring, we establish the convergence of the empirical eigenvalue distribution on the optimal local scale of order $N^{-1/2+\varepsilon}$ and establish the optimal convergence rate. The same results hold true when~$U$ and~$V$ are Haar distributed on $O(N)$.