Random graph approach to Gibbs processes with pair interaction

Abstract

This study was motivated by the observation that, in a broad class of cases, the distribution of classical Gibbs point processes in R ^d 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 graph. 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. Ambartzumian and H. S. Sukiasian [1] where the existence of a new class of stationary point processes in R ^d was established which have density (correlation) functions of the form % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaqdbaWexLMBb50ujb% qegi0BVTgib5gDPfxDHbacfiGae8NKbmOae8hkaGIae8hEaG3aaSba% a4qaaiaabgdaaeqaa0Gaaiilaiaac6cacaGGUaGaaiOlaiaacYcaca% WG4bWaaSbaa4qaaiaad6gaaeqaa0GaaiykaiaabccacqGH9aqpcaqG% GaGaamOyamaaCaaaoeqabaGaamOBaaaanmaarababaGaamiAaiaacI% cacaWG4bWaaSbaa4qaaiaadMgaaeqaaaqaaiaad6gaaeqaniabg+Gi% vdGaeyOeI0IaamiEamaaBaaaoeaacaWGQbaabeaaniaacMcaaaa!56B6!\[f(x_{\text{1}} ,...,x_n ){\text{ }} = {\text{ }}b^n \prod\nolimits_n {h(x_i } - x_j )\] (here and below, % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaqdbaWaaebaaeaada% WgaaGdbaGaamOBaaqabaaabeqab0Gaey4dIunaaaa!38CA!\[\prod {_n } \] denotes a product taken over all two-subsets {i, j} ⊄ {1,..., n}).