Hostname: page-component-76c49bb84f-t7r7g Total loading time: 0 Render date: 2025-07-09T20:10:59.889Z Has data issue: false hasContentIssue false
  • English
  • Français

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)
Get access Rights & Permissions [Opens in a new window]

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].

Information

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

[AB]Aizenmann, M. and Barsky, D. J.. Sharpness of the phase transition in percolation models. Comm. Math. Phys. 108 (1987), 489–526.CrossRefGoogle Scholar
[Be]Berger, R.. The undecidability of the Domino Problem. Mem. Amer. Math. Soc. 66 (1966), 1–72.Google Scholar
[BE]van den Berg, J. and Ermakov, A.. A new lower bound for the critical probability of site percolation on the square lattice. Random Structures Algorithms 8(3) (1996), 199–212.3.0.CO;2-T>CrossRefGoogle Scholar
[BM]van den Berg, J. and Maes, C.. Disagreement percolation in the study of Markov random fields. Ann. Probab. 22(2) (1994), 749–763.Google Scholar
[BS]van den Berg, J. and Steif, J.. Percolation and the hard-core lattice gas model. Stochastic Process. Appl. 49(2) (1994), 179–197.CrossRefGoogle Scholar
[Bl]Brascamp, H. J.. Equilibrium States for a one-dimensional lattice gas. Comm. Math. Phys. 21 (1971), 56–70.CrossRefGoogle Scholar
[GK]Gamarnik, D. and Katz, D.. Sequential cavity method for computing free energy and surface pressure. J. Stat. Phys. 137 (2009), 205–232.CrossRefGoogle Scholar
[Ge]Georgii, H.-O.. Gibbs Measures and Phase Transitions. de Gruyter, Berlin, 1988.CrossRefGoogle Scholar
[Gl]Glasner, E.. Ergodic Theory via Joinings (Mathematical Surveys and Monographs, 101). American Mathematical Society, Providence, RI, 2003.CrossRefGoogle Scholar
[Gr]Grimmett, G.. Percolation. Springer, New York, 1989.CrossRefGoogle Scholar
[HM]Hochman, M. and Meyerovitch, T.. A characterization of the entropies of multidimensional shifts of finite type. Ann. of Math. (2) 171(3) (2010), 2011–2038.CrossRefGoogle Scholar
[HorJ]Horn, R. A. and Johnson, C. P.. Matrix Analysis. Cambridge University Press, Cambridge, 1988.Google Scholar
[Ko]Ko, K.. Complexity Theory of Real Functions (Progress in Theoretical Computer Science). Birkhäuser, Boston, MA, 1991.CrossRefGoogle Scholar
[Kr]Krieger, W.. On the uniqueness of the equilibrium state. Math. Systems Theory 8(2) (1974/75), 97–104.CrossRefGoogle Scholar
[LM]Lind, D. and Marcus, B.. An Introduction to Symbolic Dynamics and Coding. Cambridge University Press, Cambridge, 1995.CrossRefGoogle Scholar
[Me]Menshikov, M. V.. Coincidence of critical points in percolation problems. Soviet Math. Dokl. 33 (1986), 856–859.Google Scholar
[Mi]Misiurewicz, M.. A short proof of the variational principle for a ℤn+-action on a compact space. Asterisque 40 (1975), 147–157.Google Scholar
[P]Pavlov, R.. Approximating the hard square entropy constant with probabilistic methods. Ann. Probab. to appear.Google Scholar
[Pi]Pierce, L.. Computing entropy for ℤ2 actions. PhD Thesis, Oregon State University, 2008.Google Scholar
[Rud]Rudolph, D. J.. Fundamentals of Measurable Dynamics. Clarendon Press, Oxford, 1990.Google Scholar
[Rue]Ruelle, D.. Thermodynamic Formalism. Addison-Wesley, Reading, MA, 1978.Google Scholar
[S]Shields, P.. The Ergodic Theory of Discrete Sample Paths (Graduate Studies in Mathematics, 13). American Mathematical Society, Providence, RI, 1996.CrossRefGoogle Scholar
[Sim]Simon, B.. The Statistical Mechanics of Lattice Gases, Volume 1 (Princeton Series in Physics). Princeton University Press, 1993.CrossRefGoogle Scholar
[Wal]Walters, P.. An Introduction to Ergodic Theory (Graduate Texts in Mathematics, 79). Springer, Berlin, 1982.CrossRefGoogle Scholar
[War]Ward, T.. Automorphisms of ℤd-subhifts of finite type. Indag. Math. (N.S.) 5(4) (1994), 495–504.CrossRefGoogle Scholar
[Wei]Weitz, D.. Computing independent sets up to the tree threshold. Proc. 38th STOC (Symposium on the Theory of Computing). ACM, New York, 2006, pp. 140–149.Google Scholar