Exponential bounds for the support convergence in the Single Ring\n Theorem

2014/09/12 by Florent Benaych-Georges, Benaych-Georges, Florent
Computer Science · Mathematics · #15B52 #46L54 #60B20 #Advanced Combinatorial Mathematics #FOS: Mathematics #Matrix Theory and Algorithms #Probability (math.PR) #Random Matrices and Applications

paper · pdf · doi:10.48550/arxiv.1409.3864

openalex publication_date 2014/09/12 · openalex created_date 2025/10/27 · openalex updated_date 2026/07/28

Abstract

We consider an n by n matrix of the form A=UTV, with U, V some\nindependent Haar-distributed unitary matrices and T a deterministic matrix.\nWe prove that for k\∼ n^1/6 and\nb²:=\(1)/(n)\Tr(|T|²), as n tends to infinity, we have\n
mathbbE
operatornameTr (A^k(A^k)^*)

lesssim
b^2k
qquad\n
textrmand
qquad
mathbbE[|
operatornameTr (A^k)|²]

lesssim
\nb^2k. This gives a simple proof (with slightly weakened hypothesis) of the\nconvergence of the support in the Single Ring Theorem, improves the available\nerror bound for this convergence from n^-\α to e^-cn^1/6 and\nproves that the rate of this convergence is at most n^-1/6\log n.\n

Related

• Support convergence in the single ring theorem
• Local single ring theorem on optimal scale
• Deformed single ring theorems
• The single ring theorem
• Strong convergence of tensor products of independent G.U.E. matrices