Random graph approach to Gibbs processes with pair interaction

✍ Scribed by R. V. Ambartzumian

• Book ID: 104622957
• Publisher: Springer Netherlands
• Year: 1991
• Tongue: English
• Weight: 521 KB
• Volume: 22
• Category: Article
• ISSN: 0167-8019
• DOI: 10.1007/bf00047648

✦ Synopsis

This study was motivated by the observation that, in a broad class of cases, the distribution of classical Gibbs point processes in R n governed by 'pair potential', can be obtained as the equilibrium distribution of a Markov chain of point processes in R d. Our analysis of this Markov chain is based on its imbedding in an infinite random 9raph. A condition of ergodicity of the chain is given in terms of the 'absence of percolation' in the graph, and this can be checked in simpler cases. The embedding also suggests a stochastic construction for the equilibrium distribution in question.

These constructions (which can also be of independent interest) are related to Gibbs processes by means of the results obtained in a recent paper of R. V. Arnbartzumian and H. S. Sukiasian where the existence of a new class of stationary point processes in R d was established which have density (correlation) functions of the formf(x 1 ..... x.) = b" l-I. h(x i -x j) (here and below, 11. denotes a product taken over all two-subsets {i,j) ~ {1 ..... n}).