A transportation approach to the mean-field approximation

The mean-field approximation is a common scheme in statistical physics to estimate the free energy of certain Gibbs measures–a key quantity on which rests many predictions of the asymptotic of the system. In this talk we will focus on rigorously justifying the mean-field approximation and obtaining quantitative bounds on the error, in particular in the case of Ising model on graphs with growing average degree. In the context of the discrete hypercube, we will see how a new transportation-entropy inequality allows us to obtain a dimension-free bound on the error induced by the mean-field approximation, relevant for the so-called Gibbs measures of low complexity.