Block Factorization of the Relative Entropy via Spatial Mixing

Abstract

We consider spin systems in the d-dimensional lattice {mathbb Z} ^d satisfying the so-called strong spatial mixing condition. We show that the relative entropy functional of the corresponding Gibbs measure satisfies a family of inequalities which control the entropy on a given region Vsubset {mathbb Z} ^d in terms of a weighted sum of the entropies on blocks Asubset V when each A is given an arbitrary nonnegative weight alpha _A. These inequalities generalize the well known logarithmic Sobolev inequality for the Glauber dynamics. Moreover, they provide a natural extension of the classical Shearer inequality satisfied by the Shannon entropy. Finally, they imply a family of modified logarithmic Sobolev inequalities which give quantitative control on the convergence to equilibrium of arbitrary weighted block dynamics of heat bath type.