Eigenvectors of Deformed Wigner Random Matrices

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
$\mathcal O({1}/{N})$
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
${1}/{\theta ^{*} - x}$
with
$\theta ^{*}:= \theta + ({1}/{\theta })$
. This distribution function will be shown to be the ordinary generating function of Chebyshev polynomials of the 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. We also extend the result to constant rank-
$r$
deformations.