Additive/multiplicative free subordination property and limiting eigenvectors of spiked additive deformations of Wigner matrices and spiked sample covariance matrices

TL;DR

This paper investigates how eigenvectors of spiked deformations of Wigner and sample covariance matrices behave asymptotically, emphasizing the role of subordination functions in free probability theory.

Contribution

It introduces a novel analysis of eigenvector projections in spiked models using free subordination properties, revealing new asymptotic behaviors.

Findings

Eigenvector projections are governed by the inverse subordination function.

Asymptotic behavior depends on the separation of eigenvalues from the bulk.

Provides new insights into eigenvector localization in spiked models.

Abstract

When some eigenvalues of a spiked multiplicative resp. additive deformation model of a Hermitian Wigner matrix resp. a sample covariance matrix separate from the bulk, we study how the corresponding eigenvectors project onto those of the perturbation. We point out that the inverse of the subordination function relative to the free additive resp. multiplicative convolution plays an important part in the asymptotic behavior.