×

zbMATH — the first resource for mathematics

Disagreement percolation for Gibbs ball models. (English) Zbl 1426.82016
Summary: We generalise disagreement percolation to Gibbs point processes of balls with varying radii. This allows to establish the uniqueness of the Gibbs measure and exponential decay of pair correlations in the low activity regime by comparison with a sub-critical Boolean model. Applications to the continuum random cluster model and the quermass-interaction model are presented. At the core of our proof lies an explicit dependent thinning from a Poisson point process to a dominated Gibbs point process.

MSC:
82B21 Continuum models (systems of particles, etc.) arising in equilibrium statistical mechanics
60E15 Inequalities; stochastic orderings
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
82B43 Percolation
60D05 Geometric probability and stochastic geometry
82B26 Phase transitions (general) in equilibrium statistical mechanics
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Ahlberg, D.; Tassion, V.; Teixeira, A., Sharpness of the phase transition for continuum percolation in R̂2, Probab. Theory Relat. Fields, 172, 525, (2018) · Zbl 1404.60143
[2] Beneš, V.; Novotná, D., Gaussian approximation for functionals of Gibbs particle processes, Kybernetika, 54, 4, 765-777, (2018) · Zbl 1449.60012
[3] van den Berg, J.; Maes, C., Disagreement percolation in the study of Markov fields, Ann. Probab., 22, 2, 749-763, (1994) · Zbl 0814.60096
[4] Bogachev, V. I., Measure Theory, Vol. I, II, (2007), Springer-Verlag: Springer-Verlag Berlin · Zbl 1120.28001
[5] Chayes, J. T.; Chayes, L.; Kotecký, R., The analysis of the Widom-Rowlinson model by stochastic geometric methods, Commn. Math. Phys., 172, 3, 551-569, (1995) · Zbl 0830.60100
[6] Coupier, D.; Dereudre, D., Continuum percolation for quermass interaction model, Electron. J. Probab., 19, 35, 19, (2014) · Zbl 1291.60201
[7] Daley, D. J.; Vere-Jones, D., (An Introduction to the Theory of Point Processes. Vol. II. An Introduction to the Theory of Point Processes. Vol. II, Probability and its Applications (New York), (2008), Springer: Springer New York), General theory and structure · Zbl 1159.60003
[8] Dereudre, D., The existence of quermass-interaction processes for nonlocally stable interaction and nonbounded convex grains, Adv. Appl. Probab., 41, 3, 664-681, (2009) · Zbl 1179.60003
[9] Dereudre, D.; Houdebert, P., Infinite volume continuum random cluster model, Electron. J. Probab., 20, 125, 1-24, (2015) · Zbl 1330.60021
[10] Dobrushin, R. L., Description of a random field by means of conditional probabilities and conditions for its regularity, Teor. Veroyatn. Primen., 13, 201-229, (1968) · Zbl 0184.40403
[11] Dobrushin, R. L.; Shlosman, S. B., Completely analytical interactions: Constructive description, J. Stat. Phys., 46, 5-6, 983-1014, (1987) · Zbl 0683.60080
[12] H. Duminil-Copin, A. Raoufi, V. Tassion, Subcritical phase of \(d\)-dimensional Poisson-Boolean percolation and its vacant set. ArXiv e-prints 1805.00695, November 2018. · Zbl 1395.82043
[13] Georgii, H.-O.; Küneth, T., Stochastic comparison of point random fields, J. Appl. Probab., 34, 4, 868-881, (1997) · Zbl 0905.60030
[14] Gouéré, J.-B., Subcritical regimes in the Poisson Boolean model of continuum percolation, Ann. Probab., 36, 4, 1209-1220, (2008) · Zbl 1148.60077
[15] Grimmett, G., (The Random-Cluster Model. The Random-Cluster Model, Grundlehren der Mathematischen Wissenschaften Fundamental Principles of Mathematical Sciences, vol. 333, (2006), Springer-Verlag: Springer-Verlag Berlin) · Zbl 1122.60087
[16] C. Hofer-Temmel, Disagreement percolation for the hard-sphere model. ArXiv e-prints 1507.02521v6, September 2017.
[17] Houdebert, P., Percolation results for the continuum random cluster model, Adv. Appl. Probab., 50, 1, 231-244, (2017)
[18] Kendall, W. S.; van Lieshout, M. N.M.; Baddeley, A. J., Quermass-interaction processes: conditions for stability, Adv. Appl. Probab., 31, 2, 315-342, (1999) · Zbl 0962.60026
[19] Klein, D., Dobrushin uniqueness techniques and the decay of correlations in continuum statistical mechanics, Commun. Math. Phys., 86, 2, 227-246, (1982) · Zbl 0503.60100
[20] Lanford, O. E.; Ruelle, D., Observables at infinity and states with short range correlations in statistical mechanics, Commun. Math. Phys., 13, 194-215, (1969)
[21] Liggett, T. M., (Interacting Particle Systems. Interacting Particle Systems, Grundlehren der Mathematischen Wissenschaften Fundamental Principles of Mathematical Sciences, vol. 276, (1985), Springer-Verlag: Springer-Verlag New York)
[22] Meester, R.; Roy, R., (Continuum Percolation. Continuum Percolation, Cambridge Tracts in Mathematics, vol. 119, (1996), Cambridge University Press: Cambridge University Press Cambridge) · Zbl 0858.60092
[23] Preston, C., (Random Fields. Random Fields, Lecture Notes in Mathematics, vol. 534, (1976), Springer-Verlag: Springer-Verlag Berlin-New York) · Zbl 0335.60074
[24] Ruelle, D., Superstable interactions in classical statistical mechanics, Commun. Math. Phys., 18, 127-159, (1970) · Zbl 0198.31101
[25] Ruelle, D., Existence of a phase transition in a continuous classical system, Phys. Rev. Lett., 27, 1040-1041, (1971)
[26] Strauss, D. J., A model for clustering, Biometrika, 62, 2, 467-475, (1975) · Zbl 0313.62044
[27] Villani, C., (Optimal transport. Optimal transport, Grundlehren der Mathematischen Wissenschaften Fundamental Principles of Mathematical Sciences, vol. 338, (2009), Springer-Verlag: Springer-Verlag Berlin), Old and new
[28] Widom, B.; Rowlinson, J. S., New model for the study of liquid-vapor phase transitions, J. Chem. Phys., 52, 1670-1684, (1970)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.