Ising Random Fields (2006)


These notes aim at giving the start of a self-contained mathematical presentation of the construction and study of Gibbs states of discrete spin systems on the lattice. For the sake of simplicity, we focus on Ising-type models. We start by describing the problem of constructing probability measures on infinite configurations, discuss the Kolmogorov approach as well as the Dobrushin approach via specifications (conditional probabilities). We then introduce the metric structure on configurations and the notion of quasilocality. The topology of local convergence on probability measures is introduced, and used to construct measures compatible with a quasilocal specification. We also discuss the Dobrushin uniqueness criterium.

Keywords: Infinite volume spin system, probability measure, specification, topology of local convergence, quasilocal specification, Dobrushin uniqueness, Ising model.




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