Hostname: page-component-76fb5796d-9pm4c Total loading time: 0 Render date: 2024-04-26T06:55:26.978Z Has data issue: false hasContentIssue false
  • English
  • Français

Descending chains, the lilypond model, and mutual-nearest-neighbour matching

Part of: Special processes Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Daryl J. Daley  and
Günter Last
Daryl J. Daley*
Affiliation:
Australian National University
Günter Last*
Affiliation:
Universität Karlsruhe
*
∗ Postal address: Mathematical Sciences Institute, Australian National University, Canberra, ACT 0200, Australia. Email address: daryl@maths.anu.edu.au
∗∗ Postal address: Institut für Mathematische Stochastik, Universität Karlsruhe, 76128 Karlsruhe, Germany. Email address: last@math.uni-karlsruhe.de
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 consider a hard-sphere model in ℝd generated by a stationary point process N and the lilypond growth protocol: at time 0, every point of N starts growing with unit speed in all directions to form a system of balls in which any particular ball ceases its growth at the instant that it collides with another ball. Some quite general conditions are given, under which it is shown that the model is well defined and exhibits no percolation. The absence of percolation is attributable to the fact that, under our assumptions, there can be no descending chains in N. The proof of this fact forms a significant part of the paper. It is also shown that, in the absence of descending chains, mutual-nearest-neighbour matching can be used to construct a bijective point map as defined by Thorisson.

Keywords

Point processstationarityhard-sphere modellilypond modelpercolationPalm probabilitydescending chain

MSC classification

Primary: 60G55: Point processes
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 37 , Issue 3 , September 2005 , pp. 604 - 628
Copyright
Copyright © Applied Probability Trust 2005 

References

Andrews, G. E. (1976). The Theory of Partitions. Addison-Wesley, London.Google Scholar
Daley, D. J. and Vere-Jones, D. (1988). An Introduction to the Theory of Point Processes. Springer, New York.Google Scholar
Daley, D. J., Stoyan, H. and Stoyan, D. (1999). The volume fraction of a Poisson germ model with maximally non-overlapping spherical grains. Adv. Appl. Prob. 31, 610–624.CrossRefGoogle Scholar
Ferrari, P. A., Landim, C. and Thorisson, H. (2002). Poisson trees, succession lines and coalescing random walks. Ann. Inst. H. Poincaré Prob. Statist. 40, 141–152.CrossRefGoogle Scholar
Häggström, O. and Meester, R. (1996). Nearest neighbor and hard sphere models in continuum percolation. Random Structures Algorithms 9, 295–315.Google Scholar
Heveling, M. and Last, G. (2005). Characterization of Palm measures via bijective point shifts. To appear in Ann. Prob. CrossRefGoogle Scholar
Heveling, M. and Last, G. (2005). Existence, uniqueness and algorithmic computation of general lilypond systems. To appear in Random Structures Algorithms.Google Scholar
Holroyd, A. E. and Peres, Y. (2003). Trees and matchings from point processes. Electronic Commun. Prob. 8, 17–27.Google Scholar
Knuth, D. E. (1998). The Art of Computer Programming, 2nd edn. Addison-Wesley, Reading, MA.Google Scholar
Mase, S. (1990). Mean characteristics of Gibbsian point processes. Ann. Inst. Statist. Math. 42, 203–220.CrossRefGoogle Scholar
Mecke, J. (1975). Invarianzeigenschaften allgemeiner Palmscher Masse. Math. Nachr. 65, 335–344.Google Scholar
Meester, R. and Roy, R. (1996). Continuum Percolation. Cambridge University Press.Google Scholar
Preston, C. J. (1976). Random Fields (Lecture Notes Math. 534). Springer, Berlin.CrossRefGoogle Scholar
Serfozo, R. (1999). Introduction to Stochastic Networks. Springer, New York.CrossRefGoogle Scholar
Stoyan, D., Kendall, W. S. and Mecke, J. (1995). Stochastic Geometry and Its Applications, 2nd edn. John Wiley, Chichester.Google Scholar
Thorisson, H. (2000). Coupling, Stationarity, and Regeneration. Springer, New York.Google Scholar