The existence of quermass-interaction processes for nonlocally stable interaction and nonbounded convex grains
Dereudre David
Source: Adv. in Appl. Probab. Volume 41, Number 3 (2009), 664-681.
Abstract
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.
Primary Subjects: 60D05, 82B21
Keywords: Stochastic geometry; Boolean model; germ--grain model; Gibbs point process; quermass-interaction process
Full-text: Access denied (no subscription detected)
We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber.
If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.aap/1253281059
Digital Object Identifier: doi:10.1239/aap/1253281059
Zentralblatt MATH identifier:
05625063
References
Baddeley, A. J. and van Lieshout, M. N. M. (1995). Area-interaction point processes. Ann. Inst. Statist. Math. 47, 601--619.
Mathematical Reviews (MathSciNet):
MR1370279
Zentralblatt MATH:
0848.60051
Digital Object Identifier: doi:10.1007/BF01856536
Georgii, H.-O. (1979). Canonical Gibbs Measure (Lecture Notes Math. 760). Springer, Berlin.
Mathematical Reviews (MathSciNet):
MR551621
Georgii, H.-O. and Häggström, O. (1996). Phase transition in continuum Potts models. Commun. Math. Phys. 181, 507--528.
Mathematical Reviews (MathSciNet):
MR1414841
Zentralblatt MATH:
0858.60098
Digital Object Identifier: doi:10.1007/BF02101013
Project Euclid: euclid.cmp/1104287771
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.
Mathematical Reviews (MathSciNet):
MR1227031
Zentralblatt MATH:
0792.60024
Digital Object Identifier: doi:10.1007/BF01192132
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.
Mathematical Reviews (MathSciNet):
MR1724554
Zentralblatt MATH:
0962.60026
Digital Object Identifier: doi:10.1239/aap/1029955137
Project Euclid: euclid.aap/1029955137
Likos, C. N., Mecke, K. R. and Wagner, H. (1995). Statistical morphology of random interface microemulsions. J. Chem. Phys. 102, 9350--9361.
Mecke, K. R. (1996). A morphological model for complex fluids. J. Phys. Condens. Matter 8, 9663--9667.
Molchanov, I. (1997). Statistics of the Boolean Model for Practitioners and Mathematicians. John Wiley, Chichester.
Zentralblatt MATH:
0878.62068
Møller, J. and Helisovà, K. (2008). Power diagrams and interaction processes for unions of discs. Adv. Appl. Prob. 40, 321--347.
Mathematical Reviews (MathSciNet):
MR2431299
Zentralblatt MATH:
1146.60322
Digital Object Identifier: doi:10.1239/aap/1214950206
Project Euclid: euclid.aap/1214950206
Nguyen, X.-X. and Zessin, H. (1979). Ergodic theorems for spatial processes. Z. Wahrscheinlichkeitsth. 48, 133--158.
Mathematical Reviews (MathSciNet):
MR534841
Preston, C. (1976). Random Fields (Lecture Notes Math. 534). Springer, Berlin.
Mathematical Reviews (MathSciNet):
MR448630
Ruelle, D. (1970). Superstable interactions in classical statistical mechanics. Commun. Math. Phys. 18, 127--159.
Mathematical Reviews (MathSciNet):
MR266565
Zentralblatt MATH:
0198.31101
Digital Object Identifier: doi:10.1007/BF01646091
Project Euclid: euclid.cmp/1103842505
Stoyan, D., Kendall, W. S. and Mecke, J. (1987). Stochastic Geometry and Its Applications. John Wiley, Chichester.
Mathematical Reviews (MathSciNet):
MR895588
Widom, B. and Rowlinson, J. S. (1970). New model for the study of liquid-vapor phase transitions. J. Chem. Phys. 52, 1670--1684.
Advances in Applied Probability
