Eigenvectors of non normal random matrices

2018/06/18 by Florent Benaych-Georges, Benaych-Georges, Florent, Ofer Zeitouni +1 · 1 citation
Mathematics · #15B52 #60B20 #FOS: Mathematics #Probability (math.PR) #math.PR #msc:15B52 #msc:60B20

paper · pdf · doi:10.48550/arxiv.1806.06806

15 pages, 1 figure. To appear in Electron. Commun. Probab

arxiv created 2018/09/26 · arxiv updated 2018/09/27

Abstract

We study the angles between the eigenvectors of a random n× n complex matrix M with density ∝ e^{-nTrV(M^*M)} and x↦ V(x²) convex. We prove that for unit eigenvectors v,v' associated with distinct eigenvalues λ,λ' that are the closest to specified points z,z' in the complex plane, the rescaled inner product √(n)(λ'-λ)\langlev,v'⟩ is uniformly sub-Gaussian, and give a more precise statement in the case of the Ginibre ensemble.

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