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Approximating entropy for a class of ℤ2 Markov random fields and pressure for a class of functions on ℤ2 shifts of finite type

Published online by Cambridge University Press:  02 February 2012

BRIAN MARCUS  and
RONNIE PAVLOV
BRIAN MARCUS
Affiliation:
Department of Mathematics, University of British Columbia, 1984 Mathematics Road, Vancouver, BC V6T 1Z2, Canada (email: marcus@math.ubc.ca, rpavlov@du.edu)
RONNIE PAVLOV
Affiliation:
Department of Mathematics, University of British Columbia, 1984 Mathematics Road, Vancouver, BC V6T 1Z2, Canada (email: marcus@math.ubc.ca, rpavlov@du.edu)
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Abstract

For a class of ℤ2 Markov Random Fields (MRFs) μ, 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 μ. These strip entropies can be computed explicitly when μ is a Gibbs state given by a nearest-neighbor interaction on a strongly irreducible nearest-neighbor ℤ2 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 ℤ2 shifts of finite type, generalizing a result in [R. Pavlov. Approximating the hard square entropy constant with probabilistic methods. Ann. Probab. to appear].

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Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 33 , Issue 1 , February 2013 , pp. 186 - 220
Copyright
Copyright © Cambridge University Press 2013

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