Outlier eigenvalues for full rank deformed single ring random matrices

2025/02/15 by Ho, Ching-Wei, Yin, Zhi, Zhong, Ping
#FOS: Mathematics #Operator Algebras (math.OA) #Probability (math.PR)

paper · doi:10.48550/arxiv.2502.10796

Abstract

Let A_n be an n × n deterministic matrix and Σ_n be a deterministic non-negative matrix such that A_n and Σ_n converge in *-moments to operators a and Σ respectively in some W^*-probability space. We consider the full rank deformed model A_n + U_n Σ_n V_n, where U_n and V_n are independent Haar-distributed random unitary matrices. In this paper, we investigate the eigenvalues of A_n + U_nΣ_n V_n in two domains that are outside the support of the Brown measure of a +u Σ. We give a sufficient condition to guarantee that outliers are stable in one domain, and we also prove that there are no outliers in the other domain. When A_n has a bounded rank, the first domain is exactly the one outside the outer boundary of the single ring, and the second domain is the inner disk of the single ring. Our results generalize the results of Benaych-Georges and Rochet (Probab. Theory Relat. Fields, 2016).

Related

• Outliers in the Single Ring Theorem
• Outliers in the spectrum of large deformed unitarily invariant models
• Outlier eigenvalues for deformed i.i.d. random matrices
• Outliers and bounded rank perturbation for non-Hermitian random band matrices
• Deformed single ring theorems