The Annals of Applied Probability

Kalikow-type decomposition for multicolor infinite range particle systems

A. Galves, N. L. Garcia, E. Löcherbach, and E. Orlandi

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Abstract

We consider a particle system on $\mathbb{Z}^{d}$ with real state space and interactions of infinite range. Assuming that the rate of change is continuous we obtain a Kalikow-type decomposition of the infinite range change rates as a mixture of finite range change rates. Furthermore, if a high noise condition holds, as an application of this decomposition, we design a feasible perfect simulation algorithm to sample from the stationary process. Finally, the perfect simulation scheme allows us to forge an algorithm to obtain an explicit construction of a coupling attaining Ornstein’s $\bar{d}$-distance for two ordered Ising probability measures.

Article information

Source
Ann. Appl. Probab. Volume 23, Number 4 (2013), 1629-1659.

Dates
First available in Project Euclid: 21 June 2013

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1371834040

Digital Object Identifier
doi:10.1214/12-AAP882

Mathematical Reviews number (MathSciNet)
MR3098444

Zentralblatt MATH identifier
1281.60079

Subjects
Primary: 60J25: Continuous-time Markov processes on general state spaces 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs

Keywords
Interacting particle systems infinite range interactions continuous spin systems perfect simulation random Markov chains Kalikow-type decomposition

Citation

Galves, A.; Garcia, N. L.; Löcherbach, E.; Orlandi, E. Kalikow-type decomposition for multicolor infinite range particle systems. Ann. Appl. Probab. 23 (2013), no. 4, 1629--1659. doi:10.1214/12-AAP882. https://projecteuclid.org/euclid.aoap/1371834040


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References

  • Bertein, F. and Galves, A. (1977/78). Une classe de systèmes de particules stable par association. Z. Wahrsch. Verw. Gebiete 41 73–85.
  • Bramson, M. and Kalikow, S. (1993). Nonuniqueness in $g$-functions. Israel J. Math. 84 153–160.
  • Brydges, D. C. (1986). A short course on cluster expansions. In Phénomènes Critiques, Systèmes Aléatoires, ThÉories de Jauge, Part I, II (Les Houches, 1984) 129–183. North-Holland, Amsterdam.
  • Cai, Y. (2005). A non-monotone CFTP perfect simulation method. Statist. Sinica 15 927–943.
  • Cessac, B., Nasser, H. and Vasquez, J. C. (2010). Spike trains statistics in integrate and fire models: Exact results. In Proceedings of the Cinquième Conférence Plénière Française de Neurosciences Computationnelles, “Neurocomp’10, Lyon, France.
  • Cessac, B., Rostro, H., Vasquez, J. C. and Viéville, T. (2009). How Gibbs distributions may naturally arise from synaptic adaptation mechanisms. A model-based argumentation. J. Stat. Phys. 136 565–602.
  • Comets, F. (1992). On consistency of a class of estimators for exponential families of Markov random fields on the lattice. Ann. Statist. 20 455–468.
  • Comets, F., Fernández, R. and Ferrari, P. A. (2002). Processes with long memory: Regenerative construction and perfect simulation. Ann. Appl. Probab. 12 921–943.
  • Comets, F. and Gidas, B. (1992). Parameter estimation for Gibbs distributions from partially observed data. Ann. Appl. Probab. 2 142–170.
  • Connor, S. B. and Kendall, W. S. (2007). Perfect simulation for a class of positive recurrent Markov chains. Ann. Appl. Probab. 17 781–808.
  • Dobrushin, R. L. (1996a). Estimates of semi-invariants for the Ising model at low temperatures. In Topics in Statistical and Theoretical Physics. Amer. Math. Soc. Transl. Ser. 2 177 59–81. Amer. Math. Soc., Providence, RI.
  • Dobrushin, R. L. (1996b). Perturbation methods of the theory of Gibbsian fields. In Lectures on Probability Theory and Statistics (Saint-Flour, 1994). Lecture Notes in Math. 1648 1–66. Springer, Berlin.
  • Fernández, P. J., Ferrari, P. A. and Grynberg, S. P. (2007). Perfectly random sampling of truncated multinormal distributions. Adv. in Appl. Probab. 39 973–990.
  • Ferrari, P. A. (1990). Ergodicity for spin systems with stirrings. Ann. Probab. 18 1523–1538.
  • Ferrari, P. A., Fernández, R. and Garcia, N. L. (2002). Perfect simulation for interacting point processes, loss networks and Ising models. Stochastic Process. Appl. 102 63–88.
  • Ferrari, P. A. and Grynberg, S. P. (2008). No phase transition for Gaussian fields with bounded spins. J. Stat. Phys. 130 195–202.
  • Ferrari, P. A., Maass, A., Martínez, S. and Ney, P. (2000). Cesàro mean distribution of group automata starting from measures with summable decay. Ergodic Theory Dynam. Systems 20 1657–1670.
  • Fill, J. A. (1998). An interruptible algorithm for perfect sampling via Markov chains. Ann. Appl. Probab. 8 131–162.
  • Gaetan, C. and Guyon, X. (2010). Spatial Statistics and Modeling. Springer, New York.
  • Galves, A., Garcia, N. L. and Prieur, C. (2010). Perfect simulation of a coupling achieving the $\overline{d}$-distance between ordered pairs of binary chains of infinite order. J. Stat. Phys. 141 669–682.
  • Galves, A., Löcherbach, E. and Orlandi, E. (2010). Perfect simulation of infinite range Gibbs measures and coupling with their finite range approximations. J. Stat. Phys. 138 476–495.
  • Garcia, N. L. and Marić, N. (2006). Existence and perfect simulation of one-dimensinal loss networks. Stochastic Process. Appl. 116 1920–1931.
  • Geyer, C. J. and Thompson, E. A. (1992). Constrained Monte Carlo maximum likelihood for dependent data. J. Roy. Statist. Soc. Ser. B 54 657–699.
  • Gibbs, A. L. (2004). Convergence in the Wasserstein metric for Markov chain Monte Carlo algorithms with applications to image restoration. Stoch. Models 20 473–492.
  • Gidas, B. (1988). Consistency of maximum likelihood and pseudolikelihood estimators for Gibbs distributions. In Stochastic Differential Systems, Stochastic Control Theory and Applications (Minneapolis, Minn., 1986). IMA Vol. Math. Appl. 10 129–145. Springer, New York.
  • Gidas, B. (1991). Parameter estimation for Gibbs distributions. I. Fully observed data. In Markov Random Fields: Theory and Applications (R. Chellapa andR. Jain, eds.). Academic, New York.
  • Huber, M. (2007). Perfect simulation for image restoration. Stoch. Models 23 475–487.
  • Janžura, M. (1997). Asymptotic results in parameter estimation for Gibbs random fields. Kybernetika (Prague) 33 133–159.
  • Kalikow, S. (1990). Random Markov processes and uniform martingales. Israel J. Math. 71 33–54.
  • Kendall, W. S. (1997). On some weighted Boolean models. In Proceedings of the International Symposium on Advances in Theory and Applications of Random Sets (Fontainebleau, 1996) 105–120. World Sci. Publ., River Edge, NJ.
  • Kendall, W. S. (1998). Perfect simulation for the area-interaction point process. In Probability Towards 2000 (New York, 1995) (L. Accardi and C. C. Heyde, eds.). Lecture Notes in Statist. 128 218–234. Springer, New York.
  • Kendall, W. (2005). Notes on perfect simulation. In Markov Chain Monte Carlo (W. S. Kendall, F. Liang and J. Wang, eds.). Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap. 7 93–146. World Sci. Publ., Hackensack, NJ.
  • Kendall, W. S. and Møller, J. (2000). Perfect simulation using dominating processes on ordered spaces, with application to locally stable point processes. Adv. in Appl. Probab. 32 844–865.
  • Kendall, W. S. and Thönnes, E. (1999). Perfect simulation in stochastic geometry. Pattern Recognition 32 1569–1586.
  • Kotecký, R. and Preiss, D. (1986). Cluster expansion for abstract polymer models. Comm. Math. Phys. 103 491–498.
  • Liggett, T. M. (1985). Interacting Particle Systems. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 276. Springer, New York.
  • Liggett, T. M. (2000). Interacting Particle Systems. Springer, Berlin.
  • Maes, C. and Shlosman, S. B. (1991). Ergodicity of probabilistic cellular automata: A constructive criterion. Comm. Math. Phys. 135 233–251.
  • Malyšev, V. A. (1980). Cluster expansions in lattice models of statistical physics and quantum field theory. Russ. Math. Surv. 35 1–62.
  • McBryan, O. A. and Spencer, T. (1977). On the decay of correlations in $\operatorname{SO}(n)$-symmetric ferromagnets. Comm. Math. Phys. 53 299–302.
  • Møller, J. and Waagepetersen, R. P. (2004). Statistical Inference and Simulation for Spatial Point Processes. Monographs on Statistics and Applied Probability 100. Chapman & Hall/CRC, Boca Raton, FL.
  • Nishimori, H. and Wong, K. Y. M. (1999). Statistical mechanics of image restoration and error-correcting codes. Phys. Rev. E 60 132–144.
  • Propp, J. G. and Wilson, D. B. (1996). Exact sampling with coupled Markov chains and applications to statistical mechanics. Random Structures and Algorithms 9 223–252.
  • Seiler, E. (1982). Gauge Theories as a Problem of Constructive Quantum Field Theory and Statistical Mechanics. Lecture Notes in Physics 159. Springer, Berlin.
  • Tanaka, K. (2002). Statistical–mechanical approach to image processing. J. Phys. A 35 R81–R150.
  • van den Berg, J. (1993). A uniqueness condition for Gibbs measures, with application to the $2$-dimensional Ising antiferromagnet. Comm. Math. Phys. 152 161–166.
  • van den Berg, J. and Maes, C. (1994). Disagreement percolation in the study of Markov fields. Ann. Probab. 22 749–763.
  • van den Berg, J. and Steif, J. E. (1999). On the existence and nonexistence of finitary codings for a class of random fields. Ann. Probab. 27 1501–1522.