  We perform a stochastic-homogenization analysis for composite materials
exhibiting a random microstructure. Under the assumptions of stationarity and
ergodicity, we characterize the Gamma-limit of a micromagnetic energy
functional defined on magnetizations taking value in the unit sphere, and
including both symmetric and antisymmetric exchange contributions. This
Gamma-limit corresponds to a micromagnetic energy functional with homogeneous
coefficients. We provide explicit formulas for the effective magnetic
properties of the composite material in terms of homogenization correctors.
Additionally, the variational analysis of the two exchange energy terms is
performed in the more general setting of functionals defined on manifold-valued
maps with Sobolev regularity, in the case in which the target manifold is a
bounded, orientable smooth surface with tubular neighborhood of uniform
thickness. Eventually, we present an explicit characterization of minimizers of
the effective exchange in the case of magnetic multilayers, providing
quantitative evidence of Dzyaloshinskii's predictions on the emergence of
helical structures in composite ferromagnetic materials with stochastic
microstructure.
