Approximating entropy for a class of ℤ2Markov random fields and pressure for a class of functions on ℤ2shifts of finite type

Abstract

AbstractFor a class of ℤ2Markov Random Fields (MRFs)μ, we show that the sequence of successive differences of entropies of induced MRFs on strips of heightnconverges exponentially fast (inn) to the entropy ofμ. These strip entropies can be computed explicitly whenμis a Gibbs state given by a nearest-neighbor interaction on a strongly irreducible nearest-neighbor ℤ2shift of finite typeX. We state this result in terms of approximations to the (topological) pressures of certain functions on such anX, 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 ℤ2shifts of finite type, generalizing a result in [R. Pavlov. Approximating the hard square entropy constant with probabilistic methods.Ann. Probab.to appear].