Exponential bounds for the support convergence in the Single Ring Theorem

We consider an n×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∼n1/6 and b2:=1nTr(|T|2), as n tends to infinity, we haveETr(Ak(Ak)⁎)≲b2k and E[|Tr(Ak)|2]≲b2k. 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−cn1/6 and proves that the rate of this convergence is at most n−1/6log⁡n.