Propagating beliefs in spin glass models

TL;DR

This paper analyzes the belief propagation algorithm's dynamics in spin glass models, revealing that its stationary states align with the replica symmetric solution and establishing a link between algorithmic stability and replica symmetry breaking.

Contribution

It provides a recursive framework for understanding BP dynamics in spin glasses and derives an instability condition matching replica symmetry breaking criteria.

Findings

BP stationary states match the replica symmetric solution.

Instability condition for BP fixed point aligns with replica symmetry breaking.

Phase diagram for finite connectivity spin glasses is numerically supported.

Abstract

We investigate dynamics of an inference algorithm termed the belief propagation (BP) when employed in spin glass (SG) models and show that its macroscopic behaviors can be traced by recursive updates of certain auxiliary field distributions whose stationary state reproduces the replica symmetric solution offered by the equilibrium analysis. We further provide a compact expression for the instability condition of the BP's fixed point which turns out to be identical to that of instability for breaking the replica symmetry in equilibrium when the number of couplings per spin is infinite. This correspondence is extended to a SG model of finite connectivity to determine the phase diagram, which is numerically supported.