Random Field Ising Model in two dimensions: Bethe approximation, Cluster Variational Method and message passing algorithms

TL;DR

This paper compares Bethe and plaquette-CVM free energy approximations for the 2D Random Field Ising Model, analyzing their accuracy and the structure of message passing algorithms, revealing limitations in defining a robust critical line.

Contribution

It introduces a comparative analysis of Bethe and plaquette-CVM methods for the 2D RFIM and explores message passing fixed points, highlighting the limitations of current approximations.

Findings

Plaquette-CVM predicts a lower, more accurate critical line.

Both methods show long-range order at low temperatures and fields.

Message passing algorithms exhibit complex fixed point structures.

Abstract

We study two free energy approximations (Bethe and plaquette-CVM) for the Random Field Ising Model in two dimensions. We compare results obtained by these two methods in single instances of the model on the square grid, showing the difficulties arising in defining a robust critical line. We also attempt average case calculations using a replica-symmetric ansatz, and compare the results with single instances. Both, Bethe and plaquette-CVM approximations present a similar panorama in the phase space, predicting long range order at low temperatures and fields. We show that plaquette-CVM is more precise, in the sense that predicts a lower critical line (the truth being no line at all). Furthermore, we give some insight on the non-trivial structure of the fixed points of different message passing algorithms.