The single ring theorem

By Alice Guionnet, Manjunath Krishnapur, Ofer Zeitouni. 2009

162 citations recorded by the source corpus.

We study the empirical measure LAn of the eigenvalues of non-normal square matrices of the form An = UnTnVn with Un;Vn independent Haar distributed on the unitary group and Tn real diagonal. We show that when the empirical measure of the eigenvalues of Tn converges, and Tn satisfies some technical conditions, LAn converges towards a rotationally invariant measure µ on the complex plane whose support is a single ring. In particular, we provide a complete proof of Feinberg-Zee single ring theorem [6]. We also consider the case where Un;Vn are independent Haar distributed on the orthogonal group.

Subfields: Algebra and Number Theory, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/8e6dd3e33a0132525bf9aa9e10a77678b7f42b80

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

In Mathematical Physics

In Statistics and Probability

Citations in the corpus

Outlier Eigenvalues for Deformed I.I.D. Random Matrices (2014, background): “The limiting empirical eigenvalues distribution of such a model is described by the so-called single ring theorem, see [30, 31, 48], and its support is of the form {z, a ≤ |z| ≤ b}.”
The norm of polynomials in large random and deterministic matrices (2010, methodology): “We use a version due to [ 18 ] which is based on the use of local concentration inequalities.”
Free Probability and Random Matrices (2014, unknown): “) This has been proved recently by Guionnet, Krishnapur, and Zeitouni [82].”
Eigenvalues of non-hermitian random matrices and Brown measure of non-normal operators: hermitian reduction and linearization method (2015, background): “That this Brown measure is actually the limit of the eigenvalue distributions of thecorresponding random matrix models was recently proved by Guionnet, Krishnapur, and Zeitouni [15].”
A Big Data Architecture Design for Smart Grids Based on Random Matrix Theory (2015, background): “3) Ring Law: Ring law for (large) non-Hermitian matrices is one of the most remarkable developments in the modern probability [24], [25].”
On the spectrum of sum and product of non-hermitian random matrices (2010, result): “The following result is due to Dozier and Silverstein.”
Invertibility of Sparse non-Hermitian matrices (2015, background): “ singular values, especially the one on the smallest singular value, play a key role in obtaining the limiting spectral distribution of various non-Hermitian random matrix ensembles. For example, see [3, 6, 10, 11, 18, 24, 30, 35]. Likewise, the bounds on the smallest singular value obtained here pl”
On the singularity of adjacency matrices for random regular digraphs (2014, background): “21) had implications for the Single Ring Theorem, proved by Guionnet, Krishnapur and Zeitouni in [17], for the limiting spectral distribution of certain random matrices with prescribed singular values; specifically, it was shown that a hypothesis in [17] could be disposed of.”