Poissonian Behavior of Ising Spin Systems in an External Field
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We apply the Stein–Chen method for Poisson approximation to spin-half Ising-type models in positive external field which satisfy the FKG inequality. In particular, we show that, provided the density of minus spins is low and can be expanded as a convergent power series in the activity (fugacity) variable, the distribution of minus spins is approximately Poisson. The error of the approximation is inversely proportional to the number of lattice sites (we obtain upper and lower bounds on the total variation distance between the exact distribution and its Poisson approximation). We illustrate these results by application to specific models, including the mean-field and nearest neighbor ferromagnetic Ising models.
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- 1.A. D. Barbour, L. Holst, and S. Janson, Poisson Approximation (Clarendon Press, Oxford, 1992).Google Scholar
- 2.K. Jacobs, S. Herminghaus, and K. R. Mecke, Thin liquid polymer films rupture via defects, Langmuir 14:965 (1998).Google Scholar
- 3.S. Herminghaus, K. Jacobs, K. R. Mecke, J. Bischof, A. Fery, M. Ibn-Elhaj, and S. Schlagowski, Spinodal dewetting in liquid and metal films, Science 282:916 (1998).Google Scholar
- 4.A. D. Barbour and P. E. Greenwood, Rates of Poisson approximation to finite range ran-dom fields, Ann. Appl. Prob. 3:91 (1993).Google Scholar
- 5.R. Ferné ndez, P. A. Ferrari, and N. L. Garcia, Loss network representation of Peierls contours. Preprint. Los Alamos Archive Math. PR/9806131, 1998.Google Scholar
- 6.C. M. Fortuin, P. W. Kasteleyn, and J. Ginibre, Correlation inequalities in some partially ordered sets, Comm. Math. Phys. 22:89 (1971).Google Scholar
- 7.E. C. Titchmarsh, The Theory of Functions, 2nd ed. (Oxford University Press, London, 1939).Google Scholar
- 8.D. Ruelle, Statistical Mechanics: Rigorous Results (Benjamin, New York, 1969).Google Scholar
- 9.C. N. Yang and T. D. Lee, Statistical theory of equations of state and phase transitions, I. Theory of condensation, Phys. Rev. 87:404 (1952).Google Scholar
- 10.T. D. Lee and C. N. Yang, Statistical theory of equations of state and phase transitions, II. Lattice gas and Ising model, Phys. Rev. 87:410 (1952).Google Scholar
- 11.S. Katsura, Phase transition of Husimi–Temperley model of imperfect gas, Prog. Theo. Phys. 13:571 (1955).Google Scholar