Partition Function Expansion on Region-Graphs and Message-Passing Equations

TL;DR

This paper develops a rigorous statistical mechanics framework for finite-dimensional disordered systems using partition function expansion and region-graphs, deriving free energy expressions and message-passing equations.

Contribution

It introduces a novel theoretical approach that extends mean-field methods to finite-dimensional systems with complex local correlations.

Findings

Derived rigorous expressions for free energy and grand free energy.

Established message-passing equations like belief-propagation on region-graphs.

Provided a mathematical foundation for analyzing finite-dimensional disordered systems.

Abstract

Disordered and frustrated graphical systems are ubiquitous in physics, biology, and information science. For models on complete graphs or random graphs, deep understanding has been achieved through the mean-field replica and cavity methods. But finite-dimensional `real' systems persist to be very challenging because of the abundance of short loops and strong local correlations. A statistical mechanics theory is constructed in this paper for finite-dimensional models based on the mathematical framework of partition function expansion and the concept of region-graphs. Rigorous expressions for the free energy and grand free energy are derived. Message-passing equations on the region-graph, such as belief-propagation and survey-propagation, are also derived rigorously.