Skip to main content Accessibility help
×
Home

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

  • Christoph Hofer-Temmel (a1)

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).

Copyright

Corresponding author

* Current address: , c/o FMW, MPC 10A, Postbus 10000, 1780 CA Den Helder, The Netherlands. Email address: math@temmel.me

References

Hide All
[1] Aaronson, J.,Gilat, D. and Keane, M. (1992).On the structure of 1-dependent Markov chains.J. Theoret. Prob. 5,545561.
[2] Aaronson, J.,Gilat, D.,Keane, M. and de Valk, V. (1989).An algebraic construction of a class of one-dependent processes.Ann. Prob. 17,128143.
[3] Alon, N. and Spencer, J. H. (2008).The Probabilistic Method,3rd edn.John Wiley,Hoboken, NJ.
[4] Baddeley, A. J.,van Lieshout, M. N. M. and Møller, J. (1996).Markov properties of cluster processes.Adv. Appl. Prob. 28,346355.
[5] Błaszczyszyn, B. and Yogeshwaran, D. (2014).On comparison of clustering properties of point processes.Adv. Appl. Prob. 46,120.
[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.3171.
[7] Borodin, A. (2011).Determinantal point processes. In The Oxford Handbook of Random Matrix Theory,Oxford University Press,pp.231249.
[8] Broman, E. I. (2005).One-dependent trigonometric determinantal processes are two-block-factors.Ann. Prob. 33,601609.
[9] Burton, R. M.,Goulet, M. and Meester, R. (1993).On 1-dependent processes and k-block factors.Ann. Prob. 21,21572168.
[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.
[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.
[12] De Valk, V. (1988).The maximal and minimal 2-correlation of a class of 1-dependent 0-1 valued processes.Israel J. Math. 62,181205.
[13] De Valk, V. (1993).Hilbert space representations of m-dependent processes.Ann. Prob. 21,15501570.
[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.5981.
[15] Eisenbaum, N. (2012).Stochastic order for alpha-permanental point processes.Stoch. Process. Appl. 122,952967.
[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.609627.
[17] Fernández, R. and Procacci, A. (2007).Cluster expansion for abstract polymer models. New bounds from an old approach.Commun. Math. Phys. 274,123140.
[18] Fernández, R.,Procacci, A. and Scoppola, B. (2007).The analyticity region of the hard sphere gas. Improved bounds.J. Statist. Phys. 128,11391143.
[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.
[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.
[21] Holroyd, A. E. (2014).One-dependent coloring by finitary factors.Preprint. Available at http://arxiv.org/abs/1411.1463v1.
[22] Janson, S. (1984).Runs in m-dependent sequences.Ann. Prob. 12,805818.
[23] Jensen, T. R. and Toft, B. (1995).Graph Coloring Problems.John Wiley,New York.
[24] Kotecký, R. and Preiss, D. (1986).Cluster expansion for abstract polymer models.Commun. Math. Phys. 103,491498.
[25] Liggett, T. M.,Schonmann, R. H. and Stacey, A. M. (1997).Domination by product measures.Ann. Prob. 25,7195.
[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.
[27] Mathieu, P. and Temmel, C. (2012). k-independent percolation on trees.Stoch. Process. Appl. 122,11291153.
[28] Miracle-Solé, S. (2010).On the theory of cluster expansions.Markov Process. Relat. Fields 16,287294.
[29] Penrose, O. (1967).Convergence of fugacity expansions for classical systems.In Statistical Mechanics: Foundations and Applications,ed. T. Bak,Benjamin,New York,pp.101109.
[30] Rolski, T. and Szekli, R. (1991).Stochastic ordering and thinning of point processes.Stoch. Process. Appl. 37,299312.
[31] Ruelle, D. (1969).Statistical Mechanics: Rigorous Results.Benjamin,New York.
[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. 118,11511261.
[33] Shearer, J. B. (1985).On a problem of Spencer.Combinatorica 5,241245.
[34] Soshnikov, A. (2000).Determinantal random point fields.Uspekhi Mat. Nauk 55,107160.
[35] Stoyan, D. and Stoyan, H. (1985).On one of Matérn's hard-core point process models.Math. Nachr. 122,205214.
[36] Teichmann, J.,Ballani, F. and van den Boogaart, K. (2013).Generalizations of Matérn's hard-core point processes.Spatial Statist. 3,3353.
[37] Temmel, C. (2014).Shearer's measure and stochastic domination of product measures.J. Theoret. Prob. 27,2240.
[38] van Lieshout, M. N. M. and Baddeley, A. J. (1996).A nonparametric measure of spatial interaction in point patterns.Statist. Neerlandica 50,344361.

Keywords

MSC classification

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

  • Christoph Hofer-Temmel (a1)

Metrics

Altmetric attention score

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.