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Physical nearest-neighbour models and non-linear time-series: III Non-zero means and sub-critical temperatures
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- Journal:
- Journal of Applied Probability / Volume 11 / Issue 4 / December 1974
- Published online by Cambridge University Press:
- 14 July 2016, pp. 715-725
- Print publication:
- December 1974
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- Article
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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
