  We examine in detail the theoretical foundations of striking long-range
couplings emerging in arrays of fluid cells connected by narrow channels by
using a lattice gas (Ising model) description of a system. We present a
reexamination of the well known exact determination of the two-point
correlation function along the edge of a channel using the transfer matrix
technique and a new interpretation is provided. The explicit form of the
correlation length is found to grow exponentially with the cross section of the
channels at the bulk two-phase coexistence. The aforementioned result is
recaptured by a refined version of the Fisher-Privman theory of first order
phase transitions in which the Boltzmann factor for a domain wall is decorated
with a contribution stemming from the point tension originated at its
endpoints. The Boltzmann factor for a domain wall together with the point
tension is then identified exactly thanks to two independent analytical
techniques, providing a critical test of the Fisher-Privman theory. We then
illustrate how to build up the network model from its elementary constituents,
the cells and the channels. Moreover, we are able to extract the strength of
the coupling between cells and express them in terms of the length and width
and coarse grained quantities such as surface and point tensions. We then
support our theoretical investigation with a series of corroborating results
based on Monte Carlo simulations. We illustrate how the long range ordering
occurs and how the latter is signaled by the thermodynamic quantities
corresponding to both planar and three-dimensional Ising arrays.
