Exponential bounds for the support convergence in the Single Ring Theorem

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 $b^2:=\frac{1}{n}\operatorname{Tr}(|T|^2)$, as $n$ tends to infinity, we have $\mathbb{E} \operatorname{Tr} (A^{k}(A^{k})^*) \ \lesssim \ b^{2k}\qquad \textrm{and} \qquad\mathbb{E}[|\operatorname{Tr} (A^{k})|^2] \ \lesssim \ b^{2k}.$ 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 $n^{-α}$ to $e^{-cn^{1/6}}$ and proves that the rate of this convergence is at most $n^{-1/6}\log n$.

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