Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-22T04:59:50.163Z Has data issue: false hasContentIssue false
  • English
  • Français

Stochastic comparison of point random fields

Part of: Equilibrium statistical mechanics Stochastic processes Special processes

Published online by Cambridge University Press:  14 July 2016

Hans-Otto Georgii  and
Torsten Küneth
Hans-Otto Georgii*
Affiliation:
Universität München
Torsten Küneth*
Affiliation:
Universität München
*
∗Postal address: Mathematisches Institut der Universität München, Theresienstr. 39, D-80333 München, Germany. E-mail: georgii@rz.mathematik.uni-muenchen.de
∗Postal address: Mathematisches Institut der Universität München, Theresienstr. 39, D-80333 München, Germany. E-mail: georgii@rz.mathematik.uni-muenchen.de
Get access Rights & Permissions [Opens in a new window]

Abstract

We give an alternative proof of a point-process version of the FKG–Holley–Preston inequality which provides a sufficient condition for stochastic domination of probability measures, and for positive correlations of increasing functions.

Keywords

STOCHASTIC DOMINATIONFKG INEQUALITYCORRELATION INEQUALITIESGIBBS DISTRIBUTIONPOISSON POINT PROCESS

MSC classification

Primary: 60G55: Point processes
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory 82B21: Continuum models (systems of particles, etc.)
Type
Research Papers
Information
Journal of Applied Probability , Volume 34 , Issue 4 , December 1997 , pp. 868 - 881
Copyright
Copyright © Applied Probability Trust 1997 

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] Baddeley, A. J. and Van Lieshout, M. N. M. (1995) Area-interaction point processes. Ann. Inst. Statist. Math. 46, 601–619.CrossRefGoogle Scholar
[2] Batty, C. J. K. and Bollmann, H. W. (1980) Generalized Holley-Preston inequalities on measure spaces and their products. Z. Wahrscheinlichkeitsth. 53, 157–174.Google Scholar
[3] Chayes, J. T., Chayes, L. and Kotecký, R. (1995) The analysis of the Widom-Rowlinson model by stochastic geometric methods. Commun. Math. Phys. 172, 551–569.Google Scholar
[4] Fortuin, C. M., Kasteleyn, P. W. and Ginibre, J. (1971) Correlation inequalities on some partially ordered sets. Commun. Math. Phys. 22, 89–103.CrossRefGoogle Scholar
[5] Georgii, H. O. and Häggström, O. (1996) Phase transitions in continuum Potts models. Commun. Math. Phys. 181, 507–528.Google Scholar
[6] Harris, T. E. (1960) A lower bound for the critical probability in a certain percolation process. Proc. Cambridge Phil. Soc. 56, 13–20.Google Scholar
[7] Holley, R. (1974) Remarks on the FKG inequalities. Commun. Math. Phys. 36, 227–232.Google Scholar
[8] Janson, S. (1984) Bounds on the distributions of extremal values of a scanning process. Stoch. Proc. Appl. 18, 313–328.Google Scholar
[9] Kallenberg, O. (1983) Random Measures. 3rd edn. Akademie, Berlin.Google Scholar
[10] Lebowitz, J. L. and Monroe, J. L. (1972) Inequalities for higher order Ising spins and continuum fluids. Commun. Math. Phys. 28, 301–311.CrossRefGoogle Scholar
[11] Lindvall, T. (1992) Lectures on the Coupling Method . Wiley, New York.Google Scholar
[12] Matthes, K., Warmuth, W. and Mecke, J. (1979) Bemerkungen zu einer Arbeit von Nguyen Xuan Xanh und Hans Zessin. Math. Nachr. 88, 117–127.CrossRefGoogle Scholar
[13] Penrose, M. D. (1991) On a continuum percolation model. Adv. Appl. Prob. 23, 536–556.Google Scholar
[14] Preston, C. J. (1974) A generalization of the FKG inequalities. Commun. Math. Phys. 36, 233–242.Google Scholar
[15] Preston, C. J. (1976) Spatial birth-and-death processes. Bull. Inst. Int. Statist. 46, 371–391.Google Scholar
[16] Ruelle, D. (1969) Statistical Mechanics. Rigorous Results . Benjamin, New York.Google Scholar
[17] Widom, B. and Rowlinson, J. S. (1970) New model for the study of liquid-vapor phase transitions. J. Chem. Phys. 52, 1670–1684.Google Scholar