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Physical nearest-neighbour models and non-linear time-series: III Non-zero means and sub-critical temperatures

Published online by Cambridge University Press:  14 July 2016

M. S. Bartlett
M. S. Bartlett*
Affiliation:
University of Oxford
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Abstract

The product moment equations previously derived are first discussed for the infinite one- and two-dimensional Ising (or autologistic) model in the case of non-zero mean, as a prelude to an examination of the probability structure in the higher-dimensional (and nominally zero mean) case below the ‘critical temperature’. Of two simple possible models, A and B, both consistent with the division of the product moment p into ergodic, and long-range non-ergodic, components, such that ρ = r (1 – m2) + m2, where r is the intrinsic correlation coefficient, it is shown that the second model B appears appropriate to the three-dimensional ‘spherical model’, but the first model A to the Ising model. Model A is defined by xi = yi + M, where M = +m or –m, and E{yi} = 0; and Model B by

Keywords

NEAREST-NEIGHBOUR LATTICE MODELSAUTO-LOGISTIC MODEL WITH NON-ZERO MEANISING MODEL AT SUB-CRITICAL TEMPERATURES
Type
Research Papers
Information
Journal of Applied Probability , Volume 11 , Issue 4 , December 1974 , pp. 715 - 725
Copyright
Copyright © Applied Probability Trust 1974 

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Footnotes

This paper was written while the author was a Visiting Fellow at the Australian National University, Canberra.

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