  We show that under a low complexity condition on the gradient of a
Hamiltonian, Gibbs distributions on the Boolean hypercube are approximate
mixtures of product measures whose probability vectors are critical points of
an associated mean-field functional. This extends a previous work by the first
author. As an application, we demonstrate how this framework helps characterize
both Ising models satisfying a mean-field condition and the conditional
distributions which arise in the emerging theory of nonlinear large deviations,
both in the dense case and in the polynomially-sparse case.
