2022/10/20 by Ching-Wei Ho, Ho, Ching-Wei, Ping Zhong +1
Mathematics · #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Operator Algebras (math.OA) #Probability (math.PR) #Random Matrices and Applications #Spectral Theory in Mathematical Physics #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.2210.11147
openalex publication_date 2022/10/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given a sequence of deterministic matrices A = AN and a sequence of deterministic nonnegative matrices Σ=ΣN such that A→ a and Σ→ σ in ∗-distribution for some operators a and σ in a finite von Neumann algebra A. Let U =UN and V=VN be independent Haar-distributed unitary matrices. We use free probability techniques to prove that, under mild assumptions, the empirical eigenvalue distribution of UΣV^*+A converges to the Brown measure of T+a, where T\inA is an R-diagonal operator freely independent from a and \vert T\vert has the same distribution as σ. The assumptions can be removed if A is Hermitian or unitary. By putting A= 0, our result removes a regularity assumption in the single ring theorem by Guionnet, Krishnapur and Zeitouni. We also prove a local convergence on optimal scale, extending the local single ring theorem of Bao, Erdős and Schnelli.