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January 2004 Mixing properties and exponential decay for lattice systems in finite volumes
Kenneth S. Alexander
Ann. Probab. 32(1A): 441-487 (January 2004). DOI: 10.1214/aop/1078415842

Abstract

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.

Citation

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Kenneth S. Alexander. "Mixing properties and exponential decay for lattice systems in finite volumes." Ann. Probab. 32 (1A) 441 - 487, January 2004. https://doi.org/10.1214/aop/1078415842

Information

Published: January 2004
First available in Project Euclid: 4 March 2004

zbMATH: 1048.60080
MathSciNet: MR2040789
Digital Object Identifier: 10.1214/aop/1078415842

Subjects:
Primary: 60K35
Secondary: 82B20

Rights: Copyright © 2004 Institute of Mathematical Statistics

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Vol.32 • No. 1A • January 2004
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