We study the discrete-to-continuum limit of ferromagnetic spin systems when the
lattice spacing tends to zero. We assume that the atoms are part of a (maybe)
nonperiodic lattice close to a flat set in a lower-dimensional space, typically a plate
in three dimensions. Scaling the particle positions by a small parameter
, we perform a
-convergence
analysis of properly rescaled interfacial-type energies. We show that, up to
subsequences, the energies converge to a surface integral defined on partitions of the
flat space. In the second part of the paper we address the issue of stochastic
homogenization in the case of random stationary lattices. A finer dependence of the
homogenized energy on the average thickness of the random lattice is analyzed for
an example of a magnetic thin system obtained by a random deposition
mechanism.
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