Local single ring theorem on optimal scale

2016/12/18 by Zhigang Bao, László Erdős, Bao, Zhigang +3 · 1 citation
Mathematics · #Random Matrices and Applications #Stochastic processes and statistical mechanics #Advanced Algebra and Geometry

paper · doi:10.48550/arxiv.1612.05920

Abstract

Let U and V be two independent N by N random matrices that are distributed according to Haar measure on U(N). Let Σ be a non-negative deterministic N by N matrix. The single ring theorem [26] asserts that the empirical eigenvalue distribution of the matrix X:= UΣV^* converges weakly, in the limit of large N, to a deterministic measure which is supported on a single ring centered at the origin in 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+ε and establish the optimal convergence rate. The same results hold true when~U and~V are Haar distributed on O(N).

Cited by

• The Single Ring Theorem and a Question of Shub

Related

• Deformed single ring theorems
• Local Single Ring Theorem
• Support convergence in the single ring theorem
• The single ring theorem
• Exponential bounds for the support convergence in the Single Ring\n Theorem