Cayley Trees and Bethe Lattices, a concise analysis for mathematicians and physicists

TL;DR

This paper critically reviews Cayley Trees and Bethe Lattices in statistical mechanics, clarifying their differences, and introduces rigorous techniques like self-similarity and Kolmogorov consistency for analyzing Bethe Lattices.

Contribution

It provides a clear, rigorous analysis of Bethe Lattices, linking mathematical techniques with physics methods like Cavity and Belief Propagation, and clarifies common misconceptions.

Findings

Clarification of differences between Cayley Trees and Bethe Lattices

Introduction of rigorous techniques for Bethe Lattices analysis

Linking Kolmogorov consistency with physics methods

Abstract

We review critically the concepts and the applications of Cayley Trees and Bethe Lattices in statistical mechanics in a tentative effort to remove widespread misuse of these simple, but yet important - and different - ideal graphs. We illustrate, in particular, two rigorous techniques to deal with Bethe Lattices, based respectively on self-similarity and on the Kolmogorov consistency theorem, linking the latter with the Cavity and Belief Propagation methods, more known to the physics community.