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In this paper we study principal components analysis in the regime of highdimensionality and high noise. Our model of the problem is a rank-onedeformation of a Wigner matrix where the signal-to-noise ratio (SNR) is ofconstant order, and we are interested in the fundamental limits of detection ofthe spike. Our main goal is to gain a fine understanding of the asymptotics forthe log-likelihood ratio process, also known as the free energy, as a functionof the SNR. Our main results are twofold. We first prove that the free energyhas a finite-size correction to its limit---the replica-symmetricformula---which we explicitly compute. This provides a formula for theKullback-Leibler divergence between the planted and null models. Second, weprove that below the reconstruction threshold, where it becomes impossible toreconstruct the spike, the log-likelihood ratio has fluctuations of constantorder and converges in distribution to a Gaussian under both the planted and(under restrictions) the null model. As a consequence, we provide a generalproof of contiguity between these two distributions that holds up to thereconstruction threshold, and is valid for an arbitrary separable prior on thespike. Formulae for the total variation distance, and the Type-I and Type-IIerrors of the optimal test are also given. Our proofs are based on Gaussianinterpolation methods and a rigorous incarnation of the cavity method, asdevised by Guerra and Talagrand in their study of the Sherrington--Kirkpatrickspin-glass model.

Finite Size Corrections and Likelihood Ratio Fluctuations in the Spiked Wigner Model