The adaptive interpolation method for proving replica formulas. Applications to the Curie-Weiss and Wigner spike models

In this contribution we give a pedagogic introduction to the newly introduced adaptive interpolation method to prove in a simple and unified way replica formulas for Bayesian optimal inference problems. Many aspects of this method can already be explained at the level of the simple Curie-Weiss spin system. This provides a new method of solution for this model which does not appear to be known. We then generalize this analysis to a paradigmatic inference problem, namely rank-one matrix estimation, also refered to as the Wigner spike model in statistics. We give many pointers to the recent literature where the method has been succesfully applied.