Approximating entropy for a class of $\zz^2$ Markov Random Fields and pressure for a class of functions on $\zz^2$ shifts of finite type

Abstract

For a class of $\zz^2$ Markov Random Fields (MRFs) $\mu$, we show that the sequence of successive differences of entropies of induced MRFs on strips of height $n$ converges exponentially fast (in $n$) to the entropy of $\mu$. These strip entropies can be computed explicitly when $\mu$ is a Gibbs state given by a nearest-neighbor interaction on a strongly irreducible nearest-neighbor $\zz^2$ shift of finite type $X$. We state this result in terms of approximations to the (topological) pressures of certain functions on such an $X$, and we show that these pressures are computable if the values taken on by the functions are computable. Finally, we show that our results apply to the hard core model and Ising model for certain parameter values of the corresponding interactions, as well as to the topological entropy of certain nearest-neighbor $\zz^2$ shifts of finite type, generalizing a result in \cite{Pa}.