For spatial exponential random graphs on Z^2, if the edge-length penalty is strong enough the finite-box Gibbs measures converge to a unique infinite-volume measure, which is exponentially mixing and satisfies a CLT, with a perfect simulation algorithm.
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Graphical Construction of Spatial Gibbs Random Graphs
For spatial exponential random graphs on Z^2, if the edge-length penalty is strong enough the finite-box Gibbs measures converge to a unique infinite-volume measure, which is exponentially mixing and satisfies a CLT, with a perfect simulation algorithm.