Message passing and Monte Carlo algorithms: connecting fixed points with metastable states

TL;DR

This paper demonstrates that Cluster Variational Method solutions at the plaquette level accurately describe metastable states in spin glasses and can predict complex statistical properties, aiding Monte Carlo simulations.

Contribution

It shows that CVM solutions at the plaquette level effectively characterize metastable states and can improve Monte Carlo methods in finite-dimensional spin glass systems.

Findings

CVM solutions match actual metastable states

Predicted overlap distributions between replicas

Message passing algorithms can accelerate Monte Carlo simulations

Abstract

Mean field-like approximations (including naive mean field, Bethe and Kikuchi and more general Cluster Variational Methods) are known to stabilize ordered phases at temperatures higher than the thermodynamical transition. For example, in the Edwards-Anderson model in 2-dimensions these approximations predict a spin glass transition at finite $T$. Here we show that the spin glass solutions of the Cluster Variational Method (CVM) at plaquette level do describe well actual metastable states of the system. Moreover, we prove that these states can be used to predict non trivial statistical quantities, like the distribution of the overlap between two replicas. Our results support the idea that message passing algorithms can be helpful to accelerate Monte Carlo simulations in finite dimensional systems.