Approach to equilibrium of Glauber dynamics in the one phase region

Abstract

We develop a new method, based on renormalization group ideas (block decimation procedure), to prove, under an assumption of strong mixing in a finite cube Λ 0 , a Logarithmic Sobolev Inequality for the Gibbs state of a discrete spin system. As a consequence we derive the hypercontractivity of the Markov semigroup of the associated Glauber dynamics and the exponential convergence to equilibrium in the uniform norm in all volumes Λ «multiples» of the cube Λ 0