Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-18T19:51:23.519Z Has data issue: false hasContentIssue false
  • English
  • Français

Asymptotic inference for an Ising lattice III. Non-zero field and ferromagnetic states

Published online by Cambridge University Press:  14 July 2016

David K. Pickard
David K. Pickard*
Affiliation:
Harvard University
*
∗Postal address: Department of Statistics, Harvard University, Science Center, One Oxford Street, Cambridge, MA 02138, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Abstract

In previous papers (1976), (1977) limit theorems were obtained for the classical Ising model in the absence of an external magnetic field, thereby providing a basis for asymptotic inference. The present paper extends these results to arbitrary external magnetic fields. Statistical inference for this model is important because its nearest-neighbour interactions provide a natural first approximation to spatial interaction among binary variables located on square lattices.

The most interesting behaviour occurs in zero field and at or beyond the critical point. In this case, the central limit result for nearest-neighbour interactions requires an unusual norming, the limiting variances may depend on the nature of the boundary conditions, and there cannot be any central limit result for external magnetic field. The implications of these phenomena for statistical inference are also discussed. In particular, the maximum likelihood estimator of magnetic field is not consistent. Rather it appears to have a non-trivial asymptotic distribution.

Keywords

ISING MODELGIBBS ENSEMBLESAUTO-LOGISTIC MODELNEAREST-NEIGHBOUR MODELSCENTRAL LIMIT THEOREMASYMPTOTIC INFERENCEMAXIMUM LIKELIHOOD
Type
Research Papers
Information
Journal of Applied Probability , Volume 16 , Issue 1 , March 1979 , pp. 12 - 24
Copyright
Copyright © Applied Probability Trust 1979 

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.)

References

Ferdinand, A. E. and Fisher, M. E. (1967) Bounded and inhomogeneous Ising models I. Phys. Rev. 185, 832–846.Google Scholar
Gallavotti, G. and Martin-Löf, A. (1975) Block spin distributions for short range attractive Ising models. Nuovo Cimento 25, 425–441.Google Scholar
Gaunt, D. S. (1974) Generalized codes and their application to Ising models with four-spin interactions including eight-vertex model. J. Phys. A 7, 1596–1612.Google Scholar
Lebowitz, J. L. (1973) Bounds on the correlations and analyticity properties of ferromagnetic Ising spin systems. Comm. Math. Phys. 33, 313–321.Google Scholar
Mccoy, B. M. and Wu, T. T. (1973) The Two Dimensional Ising Model. Harvard University Press, Cambridge, MA.Google Scholar
Martin-Löf, A. (1973) Mixing properties, differentiability of the free energy and the central limit theorem for a pure phase in the Ising model at low temperature. Comm. Math. Phys. 32, 75–92.Google Scholar
Pickard, D. K. (1976) Asymptotic inference for an Ising lattice. J. Appl. Prob. 13, 486–497.Google Scholar
Pickard, D. K. (1977) Asymptotic inference for an Ising lattice II. Adv. Appl. Prob. 9, 476–501.Google Scholar
Silvey, S. D. (1970) Statistical Inference. Penguin Books Ltd., Harmondsworth, Middlesex.Google Scholar