The Annals of Applied Probability

Two sufficient conditions for Poisson approximations in the ferromagnetic Ising model

David Coupier

Full-text: Open access

Abstract

A d-dimensional ferromagnetic Ising model on a lattice torus is considered. As the size of the lattice tends to infinity, two conditions ensuring a Poisson approximation for the distribution of the number of occurrences in the lattice of any given local configuration are suggested. The proof builds on the Stein–Chen method. The rate of the Poisson approximation and the speed of convergence to it are defined and make sense for the model. Thus, the two sufficient conditions are traduced in terms of the magnetic field and the pair potential. In particular, the Poisson approximation holds even if both potentials diverge.

Article information

Source
Ann. Appl. Probab., Volume 18, Number 4 (2008), 1326-1350.

Dates
First available in Project Euclid: 21 July 2008

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1216677124

Digital Object Identifier
doi:10.1214/1214/07-AAP487

Mathematical Reviews number (MathSciNet)
MR2434173

Zentralblatt MATH identifier
1147.82009

Subjects
Primary: 60F05: Central limit and other weak theorems
Secondary: 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs

Keywords
Poisson approximation Ising model ferromagnetic interaction Stein–Chen method

Citation

Coupier, David. Two sufficient conditions for Poisson approximations in the ferromagnetic Ising model. Ann. Appl. Probab. 18 (2008), no. 4, 1326--1350. doi:10.1214/1214/07-AAP487. https://projecteuclid.org/euclid.aoap/1216677124


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References

  • [1] Arratia, R., Goldstein, L. and Gordon, L. (1989). Two moments suffice for Poisson approximations: The Chen–Stein method. Ann. Probab. 17 9–25.
  • [2] Barbour, A. D., Holst, L. and Janson, S. (1992). Poisson Approximation. Oxford Univ. Press, New York.
  • [3] Bollobás, B. (1985). Random Graphs. Academic Press, London.
  • [4] Chazottes, J.-R. and Redig, F. (2005). Occurrence, repetition and matching of patterns in the low-temperature Ising model. J. Stat. Phys. 121 579–605.
  • [5] Chen, L. H. Y. (1975). Poisson approximation for dependent trials. Ann. Probab. 3 534–545.
  • [6] Coupier, D. (2006). Poisson approximations for the Ising model. J. Stat. Phys. 123 473–495.
  • [7] Coupier, D., Desolneux, A. and Ycart, B. (2005). Image denoising by statistical area thresholding. J. Math. Imaging Vision 22 183–197.
  • [8] Coupier, D., Doukhan, P. and Ycart, B. (2006). Zero–one laws for binary random fields. ALEA Lat. Am. J. Probab. Math. Stat. 2 157–175.
  • [9] Desolneux, A., Moisan, L. and Morel, J.-M. (2008). From Gestalt theory to image analysis. Interdisciplinary Applied Mathematics 34. Springer, New York.
  • [10] Desolneux, A., Moisan, M. and Morel, J. M. (2003). Maximal meaningful events and applications to image analysis. Ann. Statist. 31 1822–1851.
  • [11] Erdös, P. and Rényi, A. (1960). On the evolution of random graphs. Magyar Tud. Akad. Mat. Kutató Int. Közl. 5 17–60.
  • [12] Fernández, R., Ferrari, P. A. and Garcia, N. L. (2001). Loss network representation of Peierls contours. Ann. Probab. 29 902–937.
  • [13] Ferrari, P. A. and Picco, P. (2000). Poisson approximation for large-contours in low-temperature Ising models. Phys. A Statist. Mech. Appl. 279 303–311.
  • [14] Fortuin, C. M., Kasteleyn, P. W. and Ginibre, J. (1971). Correlation inequalities on some partially ordered sets. Comm. Math. Phys. 22 89–103.
  • [15] Ganesh, A., Hambly, B. M., O’Connell, N., Stark, D. and Upton, P. J. (2000). Poissonian behavior of Ising spin systems in an external field. J. Stat. Phys. 99 613–626.
  • [16] Georgii, H. O. (1988). Gibbs Measures and Phase Transitions. de Gruyter, Berlin.
  • [17] Malyshev, V. A. and Minlos, R. A. (1991). Gibbs Random Fields. Cluster Expansions. Kluwer Academic Publishers Group, Dordrecht.
  • [18] Schbath, S. (1995/97). Compound Poisson approximation of word counts in DNA sequences. ESAIM Probab. Statist. 1 1–16 (electronic).