Application of log-Sobolev inequality to the stochastic dynamics of unbounded spin systems on the lattice

Abstract We consider a ferromagnetic spin system with unbounded interactions on the $d$-dimensional integer lattice ($d \geq 1$) and investigate the convergence rate for the associated stochastic dynamics which is sometimes called the Glauber dynamics. We prove that the following two conditions are equivalent;

1. The log-Sobolev inequality for the finite volume Gibbs states holds uniformly in both the volume and the boundary condition.
2. The finite volume Glauber dynamics relaxes to equilibrium exponentially fast, uniformly in the volume whenever it starts from a tempared configuration.

The condition (1) above is known to be true, for example, when $d=1$, or the coupling constants are sufficiently small (\cite{BH98, Y97, Z96}).