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 ($d \geq 1$). We prove that the following statements for the finite volume Gibbs states are equivalent (each statement being understood to be uniform in the volume and the boundary condition),

1. The uniform log-Sobolev inequality holds,
2. The spectral gaps associated with finite volume Glauber dynamics is uniformly positive,
3. The spin-spin correlation decays exponentially.

This equivalence can be seen as an extension of the results by D. W. Stroock and B. Zegarlinski \cite{SZ92a,SZ92b} to a class of unbounded spin systems.