Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-05T04:26:46.392Z Has data issue: false hasContentIssue false
  • English
  • Français

Cluster Expansions for GIBBS Point Processes

Part of: Stochastic processes Special processes Equilibrium statistical mechanics

Published online by Cambridge University Press:  15 November 2019

S. Jansen
S. Jansen*
Affiliation:
Ludwig-Maximilians-Universität München
*
*Postal address: Mathematisches Institut, Ludwig-Maximilians Universität München, Theresienstr. 39, 80333 München, Germany.
Get access Rights & Permissions [Opens in a new window]

Abstract

We provide a sufficient condition for the uniqueness in distribution of Gibbs point processes with non-negative pairwise interaction, together with convergent expansions of the log-Laplace functional, factorial moment densities and factorial cumulant densities (correlation functions and truncated correlation functions). The criterion is a continuum version of a convergence condition by Fernández and Procacci (2007), the proof is based on the Kirkwood–Salsburg integral equations and is close in spirit to the approach by Bissacot, Fernández, and Procacci (2010). In addition, we provide formulas for cumulants of double stochastic integrals with respect to Poisson random measures (not compensated) in terms of multigraphs and pairs of partitions, explaining how to go from cluster expansions to some diagrammatic expansions (Peccati and Taqqu, 2011). We also discuss relations with generating functions for trees, branching processes, Boolean percolation and the random connection model. The presentation is self-contained and requires no preliminary knowledge of cluster expansions.

Keywords

Gibbs point processcluster expansioncumulantsbranching processBoolean percolation

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Secondary: 82B05: Classical equilibrium statistical mechanics (general) 60G55: Point processes
Type
Original Article
Information
Advances in Applied Probability , Volume 51 , Issue 4 , December 2019 , pp. 1129 - 1178
Copyright
© Applied Probability Trust 2019 

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

Baddeley, A. and Nair, G. (2012). Fast approximation of the intensity of Gibbs point processes. Electron. J. Statist. 6, 1155–1169.CrossRefGoogle Scholar
Beneš, V., Hofer-Temmel, C., Last, G. and Večeřa, J. (2019). Decorrelation of a class of Gibbs particle processes and asymptotic properties of U-statistics. Available at arXiv:1903.06553.Google Scholar
Bergeron, F., Labelle, G. and Leroux, P. (1998). Combinatorial Species and Tree-Like Structures (Encyclopaedia Math. Appl. 67). Cambridge University Press.Google Scholar
Bertini, L., Cancrini, N. and Cesi, F. (2002). The spectral gap for a Glauber-type dynamics in a continuous gas. Ann. Inst. H. Poincaré Prob. Statist. 38, 91–108.CrossRefGoogle Scholar
Bissacot, R., Fernández, R. and Procacci, A. (2010). On the convergence of cluster expansions for polymer gases. J. Statist. Phys. 139, 598–617.CrossRefGoogle Scholar
Borel, E. (1942). Sur l’emploi du théorème de Bernoulli pour faciliter le calcul d’une infinité de coefficients: application au problème de l’attente à un guichet. C. R. Acad. Sci. Paris 214, 452–456.Google Scholar
Bryc, W. (1993). A remark on the connection between the large deviation principle and the central limit theorem. Statist. Prob. Lett. 18, 253–256.CrossRefGoogle Scholar
Brydges, D. C. (1986). A short course on cluster expansions. In Phénomènes critiques, systèmes aléatoires, théories de jauge, Part I, II (Les Houches, 1984), pp. 129–183. North-Holland, Amsterdam.Google Scholar
Chiu, S. N., Stoyan, D., Kendall, W. S. and Mecke, J. (2013). Stochastic Geometry and its Applications, 3rd edn (Wiley Series in Probability and Statistics). John Wiley, Chichester.CrossRefGoogle Scholar
Conache, D., Daletskii, A., Kondratiev, Y. and Pasurek, T. (2018). Gibbs states of continuum particle systems with unbounded spins: existence and uniqueness. J. Math. Phys. 59, 1–25, 013507.CrossRefGoogle Scholar
Coniglio, A., De Angelis, U. and Forlani, A. (1977). Pair connectedness and cluster size. J. Phys. A 10, 1123.CrossRefGoogle Scholar
Daley, D. J. and Vere-Jones, D. (2003). An Introduction to the Theory of Point Processes, vol. I: Elementary Theory and Methods, 2nd edn. Springer, New York.Google Scholar
Daley, D. J. and Vere-Jones, D. (2008). An Introduction to the Theory of Point Processes, vol. II: General Theory and Structure, 2nd edn. Springer, New York.CrossRefGoogle Scholar
Dereudre, D. (2019). An introduction to the theory of Gibbsian point processes. In Stochastic Geometry (Lecture Notes Math. 2237), pp. 181–229. Springer, Cham.CrossRefGoogle Scholar
Dereudre, D., Drouilhet, R. and Georgii, H.-O. (2012). Existence of Gibbsian point processes with geometry-dependent interactions. Prob. Theory Relat. Fields 153, 643–670.CrossRefGoogle Scholar
Faris, W. G. (2010). Combinatorics and cluster expansions. Prob. Surveys 7, 157–206.CrossRefGoogle Scholar
Fernández, R. and Procacci, A. (2007). Cluster expansion for abstract polymer models: new bounds from an old approach. Commun. Math. Phys. 274, 123–140.CrossRefGoogle Scholar
Fernández, R., Ferrari, P. A. and Garcia, N. L. (2001). Loss network representation of Peierls contours. Ann. Prob. 29, 902–937.Google Scholar
Fernández, R., Groisman, P. and Saglietti, S. (2016). Stability of gas measures under perturbations and discretizations. Rev. Math. Phys. 28, 1650022.CrossRefGoogle Scholar
Fernández, R., Procacci, A. and Scoppola, B. (2007). The analyticity region of the hard sphere gas: improved bounds. J. Statist. Phys. 128, 1139–1143.CrossRefGoogle Scholar
Friedli, S. and Velenik, Y. (2018). Statistical Mechanics of Lattice Systems: A Concrete Mathematical Introduction. Cambridge University Press, Cambridge.Google Scholar
Georgii, H.-O. (1976). Canonical and grand canonical Gibbs states for continuum systems. Commun. Math. Phys. 48, 31–51.CrossRefGoogle Scholar
Georgii, H.-O. (1988). Gibbs Measures and Phase Transitions (De Gruyter Studies in Mathematics 9). Walter de Gruyter, Berlin.CrossRefGoogle Scholar
Georgii, H.-O. and Zessin, H. (1993). Large deviations and the maximum entropy principle for marked point random fields. Prob. Theory Relat. Fields 96, 177–204.CrossRefGoogle Scholar
Georgii, H.-O., Häggström, O. and Maes, C. (2001). The random geometry of equilibrium phases. Phase Transit. Crit. Phenom. 18, 1–142.CrossRefGoogle Scholar
Given, J. A. and Stell, G. (1990). The Kirkwood–Salsburg equations for random continuum percolation. J. Statist. Phys. 59, 981–1018.CrossRefGoogle Scholar
Hanke, M. (2018). Well-posedness of the iterative Boltzmann inversion. J. Statist. Phys. 170, 536–553.CrossRefGoogle Scholar
Harris, T. E. (1963). The Theory of Branching Processes (Grundlehren der Mathematischen Wissenschaften 119). Springer, Berlin.CrossRefGoogle Scholar
Hofer-Temmel, C. (2019). Disagreement percolation for the hard-sphere model. Electron. J. Prob. 24, 1–22, 91.Google Scholar
Hofer-Temmel, C. andHoudebert, P. (2019). Disagreement percolation for Gibbs ball models. Stoch. Process. Appl. 129 (10), 3922–3940.CrossRefGoogle Scholar
Jagers, P. (1975). Branching Processes with Biological Applications (Wiley Series in Probability and Mathematical Statistics). Wiley-Interscience.Google Scholar
Jansen, S. (2015). Cluster and virial expansions for the multi-species Tonks gas. J. Statist. Phys. 161, 1299–1323.CrossRefGoogle Scholar
Kondratiev, Y. G. and Kuna, T. (2003). Correlation functionals for Gibbs measures and Ruelle bounds. Methods Funct. Anal. Topology 9, 9–58.Google Scholar
Kondratiev, Y. and Lytvynov, E. (2005). Glauber dynamics of continuous particle systems. Ann. Inst. H. Poincaré Prob. Statist. 41, 685–702.CrossRefGoogle Scholar
Kondratiev, Y., Kuna, T. and Ohlerich, N. (2013). Spectral gap for Glauber type dynamics for a special class of potentials. Electron. J. Probab. 18, no. 42, 18.CrossRefGoogle Scholar
Kondratiev, Y. G., Pasurek, T. and Röckner, M. (2012). Gibbs measures of continuous systems: an analytic approach. Rev. Math. Phys. 24, 1–54, 1250026.CrossRefGoogle Scholar
Kotecký, R. and Preiss, D. (1986). Cluster expansion for abstract polymer models. Commun. Math. Phys. 103, 491–498.CrossRefGoogle Scholar
Kuna, T. (1999). Studies in configuration space analysis. Doctoral dissertation, Universität Bonn.Google Scholar
Kuna, T. and Tsagkarogiannis, D. (2018). Convergence of density expansions of correlation functions and the Ornstein–Zernike equation. Ann. Inst. H. Poincaré Prob. Statist. 19, 1115–1150.CrossRefGoogle Scholar
Last, G. (2016). Stochastic analysis for Poisson processes. In Stochastic Analysis for Poisson Point Processes (Bocconi & Springer Series: Mathematics, Statistics, Finance and Economics 7), pp. 1–36. Springer.Google Scholar
Last, G. and Penrose, M. (2018). Lectures on the Poisson Point Process (Institute of Mathematical Statistics Textbooks 7). Cambridge University Press, Cambridge.Google Scholar
Last, G. and Ziesche, S. (2017). On the Ornstein–Zernike equation for stationary cluster processes and the random connection model. Adv. Appl. Prob. 49, 1260–1287.CrossRefGoogle Scholar
Last, G., Nestmann, F. andSchulte, M. (2018). The random connection model and functions of edge-marked Poisson processes: second order properties and normal approximation. Available at arXiv:1808.01203.Google Scholar
Last, G., Penrose, M. D., Schulte, M. and Thäle, C. (2014). Moments and central limit theorems for some multivariate Poisson functionals. Adv. Appl. Prob. 46, 348–364.CrossRefGoogle Scholar
Meester, R. and Roy, R. (1996). Continuum Percolation (Camb. Tracts Math. 119). Cambridge University Press, Cambridge.CrossRefGoogle Scholar
Meester, R., Penrose, M. D. and Sarkar, A. (1997). The random connection model in high dimensions. Statist. Prob. Lett. 35, 145–153.CrossRefGoogle Scholar
Møller, J. and Waagepetersen, R. P. (2004). Statistical Inference and Simulation for Spatial Point Processes (Monogr. Statist. Appl. Prob. 100). Chapman & Hall/CRC, Boca Raton, FL.Google Scholar
Moraal, H. (1976). The Kirkwood–Salsburg equation and the virial expansion for many-body potentials. Phys. Lett. A 59, 9–10.CrossRefGoogle Scholar
Nehring, B., Poghosyan, S. and Zessin, H. (2013). On the construction of point processes in statistical mechanics. J. Math. Phys. 54, 063302.CrossRefGoogle Scholar
Nguyen, X. X. and Zessin, H. (1979). Integral and differential characterizations of the Gibbs process. Math. Nachr. 88, 105–115.Google Scholar
Peccati, G. and Taqqu, M. S. (2011). Wiener Chaos: Moments, Cumulants and Diagrams: A Survey With Computer Implementation (Bocconi & Springer Series: Mathematics, Statistics, Finance and Economics 1). Springer, Milan.CrossRefGoogle Scholar
Poghosyan, S. (2013). Gibbs distributions of quantum systems: cluster expansions and asymptotics of the partition function. Armen. J. Math. 5, 125.Google Scholar
Poghosyan, S. and Ueltschi, D. (2009). Abstract cluster expansion with applications to statistical mechanical systems. J. Math. Phys. 50, 053509.CrossRefGoogle Scholar
Preston, C. (1975). Spatial birth-and-death processes. Bull. Inst. Internat. Statist. 46, 371–391, 405–408.Google Scholar
Preston, C. (1976). Random Fields (Lecture Notes Math. 534). Springer, Berlin and New York.CrossRefGoogle Scholar
Procacci, A. and Scoppola, B. (2000). The gas phase of continuous systems of hard spheres interacting via n-body potential. Commun. Math. Phys. 211, 487–496.CrossRefGoogle Scholar
Procacci, A. and Yuhjtman, S. A. (2017). Convergence of Mayer and virial expansions and the Penrose tree-graph identity. Lett. Math. Phys. 107, 31–46.CrossRefGoogle Scholar
Rebenko, A. L. (2005). Polymer expansions for continuous classical systems with many-body interaction. Methods Funct. Anal. Topology 11, 73–87.Google Scholar
Rota, G.-C. and Wallstrom, T. C. (1997). Stochastic integrals: a combinatorial approach. Ann. Prob. 1257–1283.CrossRefGoogle Scholar
Ruelle, D. (1969). Statistical Mechanics: Rigorous Results. World Scientific.Google Scholar
Ruelle, D. (1970). Superstable interactions in classical statistical mechanics. Commun. Math. Phys. 18, 127–159.CrossRefGoogle Scholar
Schuhmacher, D. and Stucki, K. (2014). Gibbs point process approximation: total variation bounds using Stein’s method. Ann. Prob. 42, 1911–1951.CrossRefGoogle Scholar
Scott, A. D. and Sokal, A. D. (2005). The repulsive lattice gas, the independent-set polynomial, and the Lovász local lemma. J. Statist. Phys. 118, 1151–1261.CrossRefGoogle Scholar
Torquato, S. (2012). Effect of dimensionality on the continuum percolation of overlapping hyperspheres and hypercubes. J. Chem. Phys. 136, 054106.CrossRefGoogle ScholarPubMed
Ueltschi, D. (2004). Cluster expansions and correlation functions. Moscow Math. J. 4, 511–522.CrossRefGoogle Scholar
Ueltschi, D. (2017). An improved tree-graph bound. In Mini-Workshop: Cluster Expansions: From Combinatorics to Analysis Through Probability (Oberwolfach Rep. 14), eds Fernández, R., Jansen, S., and Tsagkarogiannis, D..Google Scholar
Yin, M. (2012). A cluster expansion approach to exponential random graph models. J. Stat. Mech. Theory Exp. 2012, P05004.CrossRefGoogle Scholar
Zessin, H. (2009). Der Papangelou Prozess. Izv. Nats. Akad. Nauk Armenii Mat. 44, 61–72.Google Scholar