Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-10T15:06:33.480Z Has data issue: false hasContentIssue false
  • English
  • Français

The existence of quermass-interaction processes for nonlocally stable interaction and nonbounded convex grains

Part of: Equilibrium statistical mechanics Geometric probability and stochastic geometry

Published online by Cambridge University Press:  01 July 2016

David Dereudre
David Dereudre*
Affiliation:
Université de Valenciennes et du Hainaut-Cambrésis
*
∗ Postal address: LAMAV, Université de Valenciennes et du Hainaut-Cambrésis, Le Mont Houy, 59313 Valenciennes Cedex 09, France. Email address: david.dereudre@univ-valenciennes.fr
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We prove the existence of infinite-volume quermass-interaction processes in a general setting of nonlocally stable interaction and nonbounded convex grains. No condition on the parameters of the linear combination of the Minkowski functionals is assumed. The only condition is that the square of the random radius of the grain admits exponential moments for all orders. Our methods are based on entropy and large deviation tools.

Keywords

Stochastic geometryBoolean modelgerm–grain modelGibbs point processquermass-interaction process

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry 82B21: Continuum models (systems of particles, etc.)
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 41 , Issue 3 , September 2009 , pp. 664 - 681
Copyright
Copyright © Applied Probability Trust 2009 

References

Baddeley, A. J. and van Lieshout, M. N. M. (1995). Area-interaction point processes. Ann. Inst. Statist. Math. 47, 601–619.Google Scholar
Georgii, H.-O. (1979). Canonical Gibbs Measure (Lecture Notes Math. 760). Springer, Berlin.CrossRefGoogle Scholar
Georgii, H.-O. and Häggström, O. (1996). Phase transition in continuum Potts models. Commun. Math. Phys. 181, 507–528.Google 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.Google Scholar
Kendall, W. S., van Lieshout, M. N. M. and Baddeley, A. J. (1999). Quermass-interaction processes: conditions for stability. Adv. Appl. Prob. 31, 315–342.Google Scholar
Likos, C. N., Mecke, K. R. and Wagner, H. (1995). Statistical morphology of random interface microemulsions. J. Chem. Phys. 102, 9350–9361.Google Scholar
Mecke, K. R. (1996). A morphological model for complex fluids. J. Phys. Condens. Matter 8, 9663–9667.Google Scholar
Molchanov, I. (1997). Statistics of the Boolean Model for Practitioners and Mathematicians. John Wiley, Chichester.Google Scholar
Møller, J. and Helisovà, K. (2008). Power diagrams and interaction processes for unions of discs. Adv. Appl. Prob. 40, 321–347.Google Scholar
Nguyen, X.-X. and Zessin, H. (1979). Ergodic theorems for spatial processes. Z. Wahrscheinlichkeitsth. 48, 133–158.Google Scholar
Preston, C. (1976). Random Fields (Lecture Notes Math. 534). Springer, Berlin.Google Scholar
Ruelle, D. (1970). Superstable interactions in classical statistical mechanics. Commun. Math. Phys. 18, 127–159.Google Scholar
Stoyan, D., Kendall, W. S. and Mecke, J. (1987). Stochastic Geometry and Its Applications. John Wiley, Chichester.Google Scholar
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