Block Factorization of the Relative Entropy via Spatial Mixing

We consider spin systems in thed-dimensional lattice $${\mathbb Z} ^d$$Zd 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 $$V\subset {\mathbb Z} ^d$$V⊂Zd in terms of a weighted sum of the entropies on blocks $$A\subset V$$A⊂V when eachAis given an arbitrary nonnegative weight $$\alpha _A$$αA . 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.