The single ring theorem

Abstract

We study the empirical measure $L_{A_n}$ of the eigenvalues of non-normal square matrices of the form $A_n=U_n{D}_n{V}_n$ with $U_n,V_n$ independent Haar distributed on the unitary group and $D_n$ real diagonal. We show that when the empirical measure of the eigenvalues of $D_n$ converges, and $D_n$ satisfies some technical conditions, $L_{A_n}$ converges towards a rotationally invariant measure on the complex plan whose support is a single ring. In particular, we provide a complete proof of Feinberg-Zee single ring theorem \cite{FZ}. We also consider the case where $U_n,V_n$ are independent Haar distributed on the orthogonal group.

Correction of inadequate treatment of neighborhood of z=0 in original submission, various typos corrected following referee's remarks

References in corpus (4)

• Non-Gaussian Non-Hermitean Random Matrix Theory: phase transitions and addition formalism
• On the mean density of complex eigenvalues for an ensemble of random matrices with prescribed singular values
• Random matrices: Universality of ESDs and the circular law
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Cited by in corpus (5)

• Permanental processes from products of complex and quaternionic induced Ginibre ensembles
• On the spectrum of sum and product of non-hermitian random matrices
• Support convergence in the single ring theorem
• Dimensionality Increment of PMU Data for Anomaly Detection in Low Observability Power Systems
• Convergence of the spectral measure of non normal matrices