Gibbs random graphs on point processes

Abstract

Consider a discrete locally finite subset G of Rd and the complete graph (G, E), with vertices G and edges E. We consider Gibbs measures on the set of sub-graphs with vertices G and edges E´ ⊂ E. The Gibbs interaction acts between open edges having a vertex in common. We study percolation properties of the Gibbs distribution of the graph ensemble. The main results concern percolation properties of the open edges in two cases: (a) when G is sampled from a homogeneous Poisson process; and (b) for a fixed G with sufficiently sparse points. <br />