A transportation approach to the mean-field approximation

We develop transportation-entropy inequalities which are saturated by measures such that their log-density with respect to the background measure is an affine function, in setting of uniform on discrete hypercube and exponential measure. In this sense, extends well-known result Talagrand Gaussian case. By duality, these imply a strong integrability inequality for Bernoulli processes. As result, we obtain dimension-free mean-field approximation free energy Gibbs nonlinear large deviation bound only logarithmic dependence dimension. Applied Ising model, deduce within $$O(\sqrt{n} ||J||_2)$$ energy, where n number spins $$||J||_2$$ Hilbert–Schmidt norm interaction matrix. Finally, reverse log-Sobolev similar one proved recently case Eldan Ledoux.