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

## TL;DR

This paper introduces the adaptive interpolation method, a new pedagogic approach for proving replica formulas in Bayesian inference problems, demonstrated on the Curie-Weiss model and extended to the Wigner spike model.

## Contribution

It presents the adaptive interpolation method as a simple, unified proof technique for replica formulas, including a novel solution for the Curie-Weiss model.

## Key findings

- New proof technique for replica formulas
- Application to Curie-Weiss spin system
- Extension to Wigner spike model

## Abstract

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.