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February 2001 Power-law corrections to exponential decay of connectivities and correlations in lattice models
Kenneth S. Alexander
Ann. Probab. 29(1): 92-122 (February 2001). DOI: 10.1214/aop/1008956323


Consider a translation-invariant bond percolation model on the integer lattice which has exponential decay of connectivities, that is, the probability of a connection $0 \leftrightarrow x$ by a path of open bonds decreases like $\exp\{-m(\theta)|x|\}$ for some positive constant $m(\theta)$ which may depend on the direction $\theta = x/|x|$. In two and three dimensions, it is shown that if the model has an appropriate mixing property and satisfies a special case of the FKG property, then there is at most a power-law correction to the exponential decay—there exist $A$ and $C$ such that $\exp\{-m(\theta)|x|\} \ge P(0 \leftrightarrow x) \ge A|x|^{-C} \exp\{-m(\theta)|x|\}$ for all nonzero $x$ . In four or more dimensions, a similar bound holds with $|x|^{-C}$ replaced by $\exp\{-C(\log |x|)^2\}$. In particular the power-law lower bound holds for the Fortuin-Kasteleyn random cluster model in two dimensions whenever the connectivity decays exponentially, since the mixing property is known to hold in that case. Consequently a similar bound holds for correlations in the Potts model at supercritical temperatures.


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Kenneth S. Alexander. "Power-law corrections to exponential decay of connectivities and correlations in lattice models." Ann. Probab. 29 (1) 92 - 122, February 2001.


Published: February 2001
First available in Project Euclid: 21 December 2001

zbMATH: 1034.82005
MathSciNet: MR1825143
Digital Object Identifier: 10.1214/aop/1008956323

Primary: 60K35
Secondary: 82B20 , 82B43

Keywords: Exponential decay , FK model , Ornstein-Zernike behavior , power-law correction , weak mixing

Rights: Copyright © 2001 Institute of Mathematical Statistics


Vol.29 • No. 1 • February 2001
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