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Gibbs Random Graphs

Consider a discrete locally finite subset $Γ$ of $R^d$ and the complete graph $(Γ,E)$, with vertices $Γ$ and edges $E$. We consider Gibbs measures on the set of sub-graphs with vertices $Γ$ and edges $E'\subset E$. 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 the $Γ$ is a sample from homogeneous Poisson process and (b) for a fixed $Γ$ with exponential decay of connectivity.