Chains of mean-field models

Abstract

We consider a collection of Curie–Weiss (CW) spin systems, possibly with a random field,each of which is placed along the positions of a one-dimensional chain. The CW systemsare coupled together by a Kac-type interaction in the longitudinal direction of the chainand by an infinite-range interaction in the direction transverse to the chain. Ourmotivations for studying this model come from recent findings in the theory oferror-correcting codes based on spatially coupled graphs. We find that, althoughmuch simpler than the codes, the model studied here already displays similarbehavior. We are interested in the van der Waals curve in a regime where the sizeof each Curie–Weiss model tends to infinity, and the length of the chain andrange of the Kac interaction are large but finite. Below the critical temperature,and with appropriate boundary conditions, there appears a series of equilibriumstates representing kink-like interfaces between the two equilibrium states of theindividual system. The van der Waals curve oscillates periodically around theMaxwell plateau. These oscillations have a period inversely proportional to the chainlength and an amplitude exponentially small in the range of the interaction; inother words, the spinodal points of the chain model lie exponentially close to thephase transition threshold. The amplitude of the oscillations is closely related to aPeierls–Nabarro free energy barrier for the motion of the kink along the chain. Analogiesto similar phenomena and their possible algorithmic significance for graphicalmodels of interest in coding theory and theoretical computer science are pointedout.