Eigenvectors of Deformed Wigner Random Matrices

TL;DR

This paper analyzes eigenvector behavior in rank-one deformed Wigner matrices, revealing how eigenvectors correlate with the deformation vector before the phase transition and linking this to Chebyshev polynomials.

Contribution

It provides an explicit inverse law for eigenvector alignment with the deformation vector before the phase transition, connecting it to Chebyshev polynomials and combinatorial methods.

Findings

Eigenvectors with larger eigenvalues align more strongly with the deformation vector.

The correlation distribution is related to Chebyshev polynomials of the second kind.

Most of the energy of the deformation vector concentrates in a subspace of size less than N.

Abstract

We investigate eigenvectors of rank-one deformations of random matrices $\boldsymbol B = \boldsymbol A + \theta \boldsymbol {uu}^*$ in which $\boldsymbol A \in \mathbb R^{N \times N}$ is a Wigner real symmetric random matrix, $\theta \in \mathbb R^+$, and $\boldsymbol u$ is uniformly distributed on the unit sphere. It is well known that for $\theta > 1$ the eigenvector associated with the largest eigenvalue of $\boldsymbol B$ closely estimates $\boldsymbol u$ asymptotically, while for $\theta < 1$ the eigenvectors of $\boldsymbol B$ are uninformative about $\boldsymbol u$. We examine $\mathcal O(\frac{1}{N})$ correlation of eigenvectors with $\boldsymbol u$ before phase transition and show that eigenvectors with larger eigenvalue exhibit stronger alignment with deforming vector through an explicit inverse law. This distribution function will be shown to be the ordinary generating function of Chebyshev polynomials of second kind. These polynomials form an orthogonal set with respect to the semicircle weighting function. This law is an increasing function in the support of semicircle law for eigenvalues $(-2\: ,+2)$. Therefore, most of energy of the unknown deforming vector is concentrated in a $cN$-dimensional ($c<1$) known subspace of $\boldsymbol B$. We use a combinatorial approach to prove the result.