Gibbs Random Graphs

TL;DR

This paper investigates percolation phenomena in Gibbs random graphs formed on locally finite point sets in R^d, analyzing how open edges percolate under different spatial configurations and interaction rules.

Contribution

It introduces a Gibbs measure framework for sub-graphs on point sets and studies their percolation properties in both random and fixed configurations.

Findings

Percolation occurs under certain conditions for Poisson samples.

Exponential decay of connectivity influences percolation thresholds.

Results extend understanding of phase transitions in spatial Gibbs graphs.

Abstract

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