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

We consider a hard-sphere model in ℝ d 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.

[1]  Olle Häggström,et al.  Nearest neighbor and hard sphere models in continuum percolation , 1996, Random Struct. Algorithms.

[2]  David Thomas,et al.  The Art in Computer Programming , 2001 .

[3]  T. Mattfeldt.  Stochastic Geometry and Its Applications , 1996 .

[4]  Invarianzeigenschaften allgemeiner Palmscher Ma?e , 1975 .

[5]  Poisson trees, succession lines and coalescing random walks , 2002, math/0209395.

[6]  Günter Last,et al.  Existence, uniqueness, and algorithmic computation of general lilypond systems , 2006, Random Struct. Algorithms.

[7]  S. Torquato,et al.  Random Heterogeneous Materials: Microstructure and Macroscopic Properties , 2005 .

[8]  R. Serfozo.  Introduction to Stochastic Networks , 1999 .

[9]  Matthias Heveling.  Characterization of Palm measures via bijective point-shifts , 2005 .

[10]  Torquato.  Nearest-neighbor statistics for packings of hard spheres and disks. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[11]  Emil Grosswald,et al.  The Theory of Partitions , 1984 .

[12]  Alexander E. Holroyd,et al.  Trees and Matchings from Point Processes , 2002, math/0211455.

[13]  D. Stoyan,et al.  The volume fraction of a Poisson germ model with maximally non-overlapping spherical grains , 1999, Advances in Applied Probability.

[14]  A. Cohen.  On random fields , 1967 .

[15]  S. Mase,et al.  Mean characteristics of Gibbsian point processes , 1990 .

[16]  H. Thorisson.  Coupling, stationarity, and regeneration , 2000 .