The Annals of Probability

Mixing properties and exponential decay for lattice systems in finite volumes

Kenneth S. Alexander

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An infinite-volume mixing or exponential-decay property in a spin system or percolation model reflects the inability of the influence of the configuration in one region to propagate to distant regions, but in some circumstances where such properties hold, propagation can nonetheless occur in finite volumes endowed with boundary conditions. We establish the absense [sic] of such propagation, particularly in two dimensions in finite 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 fields. For example, for the FK model in two dimensions we show that exponential decay of connectivity in infinite volume implies exponential decay in simply connected finite 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 infinite volume implies a strong mixing property in simply connected finite volumes, which includes exponential decay of correlations in simply connected finite volumes, uniformly over all such volumes and all boundary conditions.

Article information

Ann. Probab., Volume 32, Number 1A (2004), 441-487.

First available in Project Euclid: 4 March 2004

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs

Exponential decay of correlations exponential decay of connectivities FK model Potts model weak mixing strong mixing


Alexander, Kenneth S. Mixing properties and exponential decay for lattice systems in finite volumes. Ann. Probab. 32 (2004), no. 1A, 441--487. doi:10.1214/aop/1078415842.

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