Application of log-Sobolov Inequality to the Stochastic Dynamics of Unbounded Spin Systems on the Lattice

We consider a ferromagnetic lattice spin system with unbounded spins and investigate the relaxation property for the associated stochastic dynamics (the Glauber dynamics) in the finite volume case. 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 tempered configuration. This can be considered as a complementary result to the ones previously obtained for infinite volume Glauber dynamics by B. Zegarlinski (1996, Comm. Math. Phys.175, 401–432). Our result can also be viewed as an extension of the equivalence theorem known for compact spin space settings.