  In this paper we study principal components analysis in the regime of high
dimensionality and high noise. Our model of the problem is a rank-one
deformation of a Wigner matrix where the signal-to-noise ratio (SNR) is of
constant order, and we are interested in the fundamental limits of detection of
the spike. Our main goal is to gain a fine understanding of the asymptotics for
the log-likelihood ratio process, also known as the free energy, as a function
of the SNR. Our main results are twofold. We first prove that the free energy
has a finite-size correction to its limit---the replica-symmetric
formula---which we explicitly compute. This provides a formula for the
Kullback-Leibler divergence between the planted and null models. Second, we
prove that below the reconstruction threshold, where it becomes impossible to
reconstruct the spike, the log-likelihood ratio has fluctuations of constant
order and converges in distribution to a Gaussian under both the planted and
(under restrictions) the null model. As a consequence, we provide a general
proof of contiguity between these two distributions that holds up to the
reconstruction threshold, and is valid for an arbitrary separable prior on the
spike. Formulae for the total variation distance, and the Type-I and Type-II
errors of the optimal test are also given. Our proofs are based on Gaussian
interpolation methods and a rigorous incarnation of the cavity method, as
devised by Guerra and Talagrand in their study of the Sherrington--Kirkpatrick
spin-glass model.
