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.
A note on perfect simulation for exponential random graph models
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In this paper we propose a perfect simulation algorithm for the Exponential Random Graph Model, based on the Coupling From The Past method of Propp & Wilson (1996). We use a Glauber dynamics to construct the Markov Chain and we prove the monotonicity of the ERGM for a subset of the parametric space. We also obtain an upper bound on the running time of the algorithm that depends on the mixing time of the Markov chain.
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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.