A new criterion for the logarithmic Sobolev inequality and two applications

Abstract

We give a criterion for the logarithmic Sobolev inequality (LSI) on the product space X1 × ⋯ × XN. We have in mind an N-site lattice, unbounded continuous spin variables, and Glauber dynamics. The interactions are described by the Hamiltonian H of the Gibbs measure. The criterion for LSI is formulated in terms of the LSI constants of the single-site conditional measures and the size of the off-diagonal entries of the Hessian of H. It is optimal for Gaussians with positive covariance matrix. To illustrate, we give two applications: one with weak interactions and one with strong interactions and a decay of correlations condition. © 2006 Elsevier Inc. All rights reserved.