  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 systems are coupled together by a Kac-type interaction in the
longitudinal direction of the chain and by an infinite range interaction in the
direction transverse to the chain. Our motivations for studying this model come
from recent findings in the theory of error correcting codes based on spatially
coupled graphs. We find that, although much simpler than the codes, the model
studied here already displays similar behaviors. We are interested in the van
der Waals curve in a regime where the size of each Curie-Weiss model tends to
infinity, and the length of the chain and range of the Kac interaction are
large but finite. Below the critical temperature, and with appropriate boundary
conditions, there appears a series of equilibrium states representing kink-like
interfaces between the two equilibrium states of the individual system. The van
der Waals curve oscillates periodically around the Maxwell plateau. These
oscillations have a period inversely proportional to the chain length and an
amplitude exponentially small in the range of the interaction; in other words
the spinodal points of the chain model lie exponentially close to the phase
transition threshold. The amplitude of the oscillations is closely related to a
Peierls-Nabarro free energy barrier for the motion of the kink along the chain.
Analogies to similar phenomena and their possible algorithmic significance for
graphical models of interest in coding theory and theoretical computer science
are pointed out.
