Block factorization of the relative entropy via spatial mixing

Abstract

We consider spin systems in the $d$-dimensional lattice $Z^d$ 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 $V\subset Z^d$ in terms of a weighted sum of the entropies on blocks $A\subset V$ 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.

23 pages, 2 figures

References in corpus (2)

• Logarithmic Sobolev inequalities in discrete product spaces: a proof by a transportation cost distance
• Entropy and expansion

Cited by in corpus (3)

• Optimal Mixing of Glauber Dynamics: Entropy Factorization via High-Dimensional Expansion
• Optimal Mixing Time for the Ising Model in the Uniqueness Regime
• Entropy decay in the Swendsen-Wang dynamics on ${\mathbb{Z}}^d$