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Unilateral Markov fields

Published online by Cambridge University Press:  01 July 2016

David K. Pickard
David K. Pickard*
Affiliation:
Harvard University
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Abstract

Recently, there has been considerable interest in some specialized binary lattice processes. However, this exciting work has been rather fragmentary and heuristic. Rigorous proofs are provided for the existence of some classes of stationary unilateral Markov fields on infinite square lattices. (Neighbours are sites which are horizontally, vertically, or diagonally adjacent.) Many of the difficulties are avoided by characterizing stationary unilateral Markov fields on finite lattices first. Detailed analyses are given for both binary and Gaussian variables.

Keywords

MARKOV FIELDSGIBBS ENSEMBLESISING MODELSCRYSTAL GROWTH-DISORDER MODELSUNILATERAL PROCESSESCORRELATIONS EMBEDDED MARKOV CHAINS
Type
Research Article
Information
Advances in Applied Probability , Volume 12 , Issue 3 , September 1980 , pp. 655 - 671
Copyright
Copyright © Applied Probability Trust 1980 

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References

Bartlett, M. S. (1967) Inference and stochastic processes. J. R. Statist Soc. A130, 457–477.Google Scholar
Bartlett, M. S. (1971) Physical nearest neighbour models and non-linear time series. J. Appl. Prob. 8, 222–232.CrossRefGoogle Scholar
Bartlett, M. S. and Besag, J. E. (1969) Correlation properties of some nearest neighbour models. Bull. Intenat. Statist. Inst. 43, 191–193.Google Scholar
Besag, J. E. (1972) Nearest-neighbour systems and the autologistic model for binary data. J. R. Statist. Soc. B34, 73–83.Google Scholar
Besag, J. E. (1974) Spatial interaction and the statistical analysis of lattice systems. J. R. Statist. Soc. B36, 192–225.Google Scholar
Breiman, L. (1968) Probability. Addison Wesley, Reading, MA.Google Scholar
Enting, I. G. (1977) Crystal growth models and Ising models. J. Phys. C, 10, 1379–1388.Google Scholar
Enting, I. G. and Welberry, T. R. (1978) Connections between Ising models and various probability distributions. Suppl. Adv. Appl. Prob. 10, 65–72.Google Scholar
Galbraith, R. F. and Walley, D. (1976) On a two-dimensional binary process. J. Appl. Prob. 13, 548–557.CrossRefGoogle Scholar
Galbraith, R. F. and Walley, D. (1980) Ergodic properties of a two-dimensional binary process. J. Appl. Prob. 17, 124–133.Google Scholar
Hurst, C. A. and Green, H. S. (1960) New solution to the Ising problem for a rectangular lattice. J. Chem. Phys. 33, 1059–1063.Google Scholar
Kaufmann, B. and Onsager, L. (1949) Crystal Statistics III. Short range order in a binary Ising lattice. Phys. Rev. 76, 1244–1252.Google Scholar
Pickard, D. K. (1977a) A curious binary lattice process. J. Appl. Prob. 14, 717–731.CrossRefGoogle Scholar
Pickard, D. K. (1977b) Statistical Inference on Lattices. Ph.D. Thesis, The Australian National University.CrossRefGoogle Scholar
Pickard, D. K. (1978) Unilateral Ising models. Suppl. Adv. Appl. Prob. 10, 57–64.CrossRefGoogle Scholar
Preston, C. J. (1974) Gibbs States on Countable Sets. Cambridge University Press, Cambridge.Google Scholar
Verhagen, A. M. W. (1977) A three parameter isotropic distribution of atoms and the hard-core square lattice gas. J. Chem. Phys. 67, 5060–5065.Google Scholar
Welberry, T. R. (1977) Solution of crystal growth disorder models by imposition of symmetry. Proc. R. Soc. London A353, 363–376.Google Scholar
Welberry, T. R. and Galbralth, R. (1973) A two-dimensional model of crystal growth disorder. J. Appl. Cryst. 6, 87–96.CrossRefGoogle Scholar
Welberry, T. R. and Galbraith, R. (1975) The effect of non-linearity on a two-dimensional model of crystal growth disorder. J. Appl. Cryst. 8, 636–644.CrossRefGoogle Scholar