Exponential bounds for the support convergence in the Single Ring Theorem

Abstract

We consider an $n$ by $n$ matrix of the form $A=UTV$, with $U,V$ some independent Haar-distributed unitary matrices and $T$ a deterministic matrix. We prove that for $k\sim n^{1/6}$ and $b^2:=\frac{1}{n}\mathrm{Tr}(|T{|}^2)$, as $n$ tends to infinity, we have $$\mathbb{E}\mathrm{Tr}(A^k(A^k{)}^{\ast })\lesssim b^{2k}\text{and}\mathbb{E}[|\mathrm{Tr}(A^k){|}^2]\lesssim b^{2k}.$$ This gives a simple proof (with slightly weakened hypothesis) of the convergence of the support in the Single Ring Theorem, improves the available error bound for this convergence from $n^{-\alpha }$ to $e^{-c{n}^{1/6}}$ and proves that the rate of this convergence is at most $n^{-1/6}\log n$.

15 pages, 1 figure. Minor typos corrected, references added. To appear in J. Funct. Anal