Estimation in the spiked Wigner model: A short proof of the replica formula

We consider the problem of estimating the rank-one perturbation of a Wigner matrix in a setting of low signal-to-noise ratio. This serves as a simple model for principal component analysis in high dimensions. The mutual information per variable between the spike and the observed matrix, or equivalently, the normalized Kullback-Leibler divergence between the planted and null models are known to converge to the so-called replica-symmetric formula, the properties of which determine the fundamental limits of estimation in this model. We provide in this note a short and transparent proof of this formula, based on simple executions of Gaussian interpolations and standard concentration-of-measure arguments. The Franz-Parisi potential, that is, the free entropy at a fixed overlap, plays an important role in our proof. Furthermore, our proof can be generalized straightforwardly to spiked tensor models of even order.