We study the magnetic energy of magnetic polymer composite materials as the average distance between magnetic particles vanishes. We model the position of these particles in the polymeric matrix as a stochastic lattice scaled by a small parameter $\varepsilon$ and the magnets as classical $\pm 1$ spin variables interacting via an Ising type energy. Under surface scaling of the energy we prove, in terms of $\Gamma$-convergence that, up to subsequences, the (continuum) $\Gamma$-limit of these energies is finite on the set of Caccioppoli partitions representing the magnetic Weiss domains where it has a local integral structure. Assuming stationarity of the stochastic lattice, we can make use of ergodic theory to further show that the $\Gamma$-limit exists and that the integrand is given by an asymptotic homogenization formula which becomes deterministic if the lattice is ergodic.
Categories: math.AP