Local Single Ring Theorem

Abstract : The Single Ring Theorem, by Guionnet, Krishnapur and Zeitouni, describes the empirical eigenvalues distribution of a large generic matrix with prescribed singular values, i.e. an $N \times N$ matrix of the form $A=UTV$, with $U, V$ some independent Haar-distributed unitary matrices and $T$ a deterministic matrix whose singular values are the ones prescribed. In this text, we give a local version of this result, proving that it remains true at the microscopic scale $(\log N)^{-1/4}$. On our way to prove it, we prove a matrix subordination result for singular values of sums of non Hermitian matrices, as Kargin did for Hermitian matrices. This also allows to prove a local law for the singular values of the sum of two non Hermitian matrices and a delocalization result for singular vectors.
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https://hal.archives-ouvertes.fr/hal-01112454
Contributor : Florent Benaych-Georges <>
Submitted on : Tuesday, February 3, 2015 - 7:17:09 AM
Last modification on : Thursday, April 11, 2019 - 4:02:09 PM

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  • HAL Id : hal-01112454, version 1
  • ARXIV : 1501.07840

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Florent Benaych-Georges. Local Single Ring Theorem. 2015. ⟨hal-01112454⟩

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