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.
"Decomposition of mean-field Gibbs distributions into product measures." Electron. J. Probab. 23 1 - 24, 2018. https://doi.org/10.1214/18-EJP159