Eigenvector dynamics under free addition

TL;DR

This paper analyzes how a specific eigenvector of a symmetric matrix evolves when the matrix is perturbed by a Gaussian orthogonal ensemble matrix, revealing a 'Cauchy-flight' regime and providing detailed eigenvector density insights.

Contribution

It introduces a non-perturbative analysis of eigenvector dynamics under Gaussian orthogonal ensemble perturbations, including a precise characterization of the 'Cauchy-flight' regime.

Findings

Quantifies eigenvector overlap during matrix perturbation

Identifies a 'Cauchy-flight' regime in eigenvector evolution

Provides a local eigenvector density in the initial matrix's eigenvalues space

Abstract

We investigate the evolution of a given eigenvector of a symmetric (deterministic or random) matrix under the addition of a matrix in the Gaussian orthogonal ensemble. We quantify the overlap between this single vector with the eigenvectors of the initial matrix and identify precisely a "Cauchy-flight" regime. In particular, we compute the local density of this vector in the eigenvalues space of the initial matrix. Our results are obtained in a non perturbative setting and are derived using the ideas of [O. Ledoit and S. P\'ech\'e, Prob. Th. Rel. Fields, {\bf 151} 233 (2011)]. Finally, we give a robust derivation of a result obtained in [R. Allez and J.-P. Bouchaud, Phys. Rev. E {\bf 86}, 046202 (2012)] to study eigenspace dynamics in a semi-perturbative regime.