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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
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Abstract

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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 

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