Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-22T13:01:30.547Z Has data issue: false hasContentIssue false
  • English
  • Français

Shearer's point process, the hard-sphere model, and a continuum Lovász local lemma

Part of: Equilibrium statistical mechanics Graph theory Stochastic processes

Published online by Cambridge University Press:  17 March 2017

Christoph Hofer-Temmel
Christoph Hofer-Temmel*
Affiliation:
VU University Amsterdam
*
* Current address: , c/o FMW, MPC 10A, Postbus 10000, 1780 CA Den Helder, The Netherlands. Email address: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

A point process is R-dependent if it behaves independently beyond the minimum distance R. In this paper we investigate uniform positive lower bounds on the avoidance functions of R-dependent simple point processes with a common intensity. Intensities with such bounds are characterised by the existence of Shearer's point process, the unique R-dependent and R-hard-core point process with a given intensity. We also present several extensions of the Lovász local lemma, a sufficient condition on the intensity and R to guarantee the existence of Shearer's point process and exponential lower bounds. Shearer's point process shares a combinatorial structure with the hard-sphere model with radius R, the unique R-hard-core Markov point process. Bounds from the Lovász local lemma convert into lower bounds on the radius of convergence of a high-temperature cluster expansion of the hard-sphere model. This recovers a classic result of Ruelle (1969) on the uniqueness of the Gibbs measure of the hard-sphere model via an inductive approach of Dobrushin (1996).

Keywords

Avoidance functionthe Lovász local lemmahard-sphere modelpartition functionR-dependent point processR-hard-core

MSC classification

Primary: 60G55: Point processes
Secondary: 60G60: Random fields 82B05: Classical equilibrium statistical mechanics (general) 05C69: Dominating sets, independent sets, cliques
Type
Research Article
Information
Advances in Applied Probability , Volume 49 , Issue 1 , March 2017 , pp. 1 - 23
Copyright
Copyright © Applied Probability Trust 2017 

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

[1] Aaronson, J.,Gilat, D. and Keane, M. (1992).On the structure of 1-dependent Markov chains.J. Theoret. Prob. 5,545–561.CrossRefGoogle Scholar
[2] Aaronson, J.,Gilat, D.,Keane, M. and de Valk, V. (1989).An algebraic construction of a class of one-dependent processes.Ann. Prob. 17,128–143.CrossRefGoogle Scholar
[3] Alon, N. and Spencer, J. H. (2008).The Probabilistic Method,3rd edn.John Wiley,Hoboken, NJ.CrossRefGoogle Scholar
[4] Baddeley, A. J.,van Lieshout, M. N. M. and Møller, J. (1996).Markov properties of cluster processes.Adv. Appl. Prob. 28,346–355.CrossRefGoogle Scholar
[5] Błaszczyszyn, B. and Yogeshwaran, D. (2014).On comparison of clustering properties of point processes.Adv. Appl. Prob. 46,1–20.CrossRefGoogle Scholar
[6] Błaszczyszyn, B. and Yogeshwaran, D. (2015).Clustering comparison of point processes, with applications to random geometric models. In Stochastic Geometry, Spatial Statistics and Random Fields (Lecture Notes Math. 2120),Springer,Cham,pp.31–71.CrossRefGoogle Scholar
[7] Borodin, A. (2011).Determinantal point processes. In The Oxford Handbook of Random Matrix Theory,Oxford University Press,pp.231–249.Google Scholar
[8] Broman, E. I. (2005).One-dependent trigonometric determinantal processes are two-block-factors.Ann. Prob. 33,601–609.CrossRefGoogle Scholar
[9] Burton, R. M.,Goulet, M. and Meester, R. (1993).On 1-dependent processes and k-block factors.Ann. Prob. 21,2157–2168.CrossRefGoogle Scholar
[10] Daley, D. J. and Vere-Jones, D. (2003).An introduction to the Theory of Point Processes: Elementary Theory and Methods,Vol. I,2nd edn.Springer,New York.Google Scholar
[11] Daley, D. J. and Vere-Jones, D. (2008).An Introduction to the Theory of Point Processes: General Theory and Structure,Vol. II,2nd edn.Springer,New York.CrossRefGoogle Scholar
[12] De Valk, V. (1988).The maximal and minimal 2-correlation of a class of 1-dependent 0-1 valued processes.Israel J. Math. 62,181–205.CrossRefGoogle Scholar
[13] De Valk, V. (1993).Hilbert space representations of m-dependent processes.Ann. Prob. 21,1550–1570.Google Scholar
[14] Dobrushin, R. L. (1996).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),American Mathematical Society,Providence, RI,pp.59–81.Google Scholar
[15] Eisenbaum, N. (2012).Stochastic order for alpha-permanental point processes.Stoch. Process. Appl. 122,952–967.CrossRefGoogle Scholar
[16] Erdös, P. and Lovász, L. (1975).Problems and results on 3-chromatic hypergraphs and some related questions. In Infinite and Finite Sets: To Paul Erdös on His 60th Birthday(Keszthely, 1973; Colloq. Math. Soc. János Bolyai 10),Vol. II,North-Holland,Amsterdam,pp.609–627.Google Scholar
[17] 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
[18] 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
[19] Hofer-Temmel, C. (2015).Shearer's point process and the hard-sphere model in one dimension.Preprint. Available at http://arxiv.org/abs/1504.02672v1.Google Scholar
[20] Hofer-Temmel, C. and Lehner, F. (2015).Clique trees of infinite locally finite chordal graphs.Preprint. Available at http://arxiv.org/abs/1311.7001v3.Google Scholar
[21] Holroyd, A. E. (2014).One-dependent coloring by finitary factors.Preprint. Available at http://arxiv.org/abs/1411.1463v1.Google Scholar
[22] Janson, S. (1984).Runs in m-dependent sequences.Ann. Prob. 12,805–818.CrossRefGoogle Scholar
[23] Jensen, T. R. and Toft, B. (1995).Graph Coloring Problems.John Wiley,New York.Google Scholar
[24] Kotecký, R. and Preiss, D. (1986).Cluster expansion for abstract polymer models.Commun. Math. Phys. 103,491–498.CrossRefGoogle Scholar
[25] Liggett, T. M.,Schonmann, R. H. and Stacey, A. M. (1997).Domination by product measures.Ann. Prob. 25,71–95.CrossRefGoogle Scholar
[26] Matérn, B. (1960).Spatial Variation: Stochastic Models and Their Application to Some Problems in Forest Surveys and Other Sampling Investigations.Meddelanden Från Statens Skogsforskningsinstitut 49,Stockholm.Google Scholar
[27] Mathieu, P. and Temmel, C. (2012). k-independent percolation on trees.Stoch. Process. Appl. 122,1129–1153.CrossRefGoogle Scholar
[28] Miracle-Solé, S. (2010).On the theory of cluster expansions.Markov Process. Relat. Fields 16,287–294.Google Scholar
[29] Penrose, O. (1967).Convergence of fugacity expansions for classical systems.In Statistical Mechanics: Foundations and Applications,ed. T. Bak,Benjamin,New York,pp.101–109.Google Scholar
[30] Rolski, T. and Szekli, R. (1991).Stochastic ordering and thinning of point processes.Stoch. Process. Appl. 37,299–312.CrossRefGoogle Scholar
[31] Ruelle, D. (1969).Statistical Mechanics: Rigorous Results.Benjamin,New York.Google Scholar
[32] 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
[33] Shearer, J. B. (1985).On a problem of Spencer.Combinatorica 5,241–245.CrossRefGoogle Scholar
[34] Soshnikov, A. (2000).Determinantal random point fields.Uspekhi Mat. Nauk 55,107–160.Google Scholar
[35] Stoyan, D. and Stoyan, H. (1985).On one of Matérn's hard-core point process models.Math. Nachr. 122,205–214.CrossRefGoogle Scholar
[36] Teichmann, J.,Ballani, F. and van den Boogaart, K. (2013).Generalizations of Matérn's hard-core point processes.Spatial Statist. 3,33–53.CrossRefGoogle Scholar
[37] Temmel, C. (2014).Shearer's measure and stochastic domination of product measures.J. Theoret. Prob. 27,22–40.CrossRefGoogle Scholar
[38] van Lieshout, M. N. M. and Baddeley, A. J. (1996).A nonparametric measure of spatial interaction in point patterns.Statist. Neerlandica 50,344–361.CrossRefGoogle Scholar