How to Compute Loop Corrections to Bethe Approximation

TL;DR

This paper presents a novel method for calculating loop corrections to the Bethe approximation in spin models, accounting for fluctuations at all scales, and demonstrates its application to Ising models and spin glasses.

Contribution

The paper introduces a new approach for computing corrections to Bethe approximation that considers all fluctuation scales, improving upon cluster variational methods.

Findings

Re-derivation of the Ginzburg criterion and upper critical dimension for the Ising model

Calculation of finite-size corrections to free energy in spin glasses

Application of the method to high-temperature phases of spin models

Abstract

We introduce a method for computing corrections to Bethe approximation for spin models on arbitrary lattices. Unlike cluster variational methods, the new approach takes into account fluctuations on all length scales. The derivation of the leading correction is explained and applied to two simple examples: the ferromagnetic Ising model on d-dimensional lattices, and the spin glass on random graphs (both in their high-temperature phases). In the first case we rederive the well-known Ginzburg criterion and the upper critical dimension. In the second, we compute finite-size corrections to the free energy.