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

Abstract

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.