Abstract:
Abstract. An innite-volume mixing or exponential-decay property in a spin system or percolation model re
ects the inability of the in
uence of the conguration in one region to propagate to distant regions, but in some circumstances where such properties hold, propagation can nonetheless occur in nite volumes endowed with boundary conditions. We establish the absense of such propagation, particularly in two dimensions in nite volumes which are simply connected, under a variety of conditions, mainly for the Potts model and the Fortuin-Kasteleyn (FK) random cluster model, allowing external elds. For example, for the FK model in two dimensions we show that exponential decay of connectivity in innite volume implies exponential decay in simply connected nite volumes, uniformly over all such volumes and all boundary conditions, and implies a strong mixing property for such volumes with certain types of boundary conditions. For the Potts model in two dimensions we show that exponential decay of correlations in innite volume implies a strong mixing property in simply connected nite volumes, which includes exponential decay of correlations in simply connected nite volumes, uniformly over all such volumes and all boundary conditions.
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