The Annals of Probability

Power-law corrections to exponential decay of connectivities and correlations in lattice models

Kenneth S. Alexander

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Abstract

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.

Article information

Source
Ann. Probab., Volume 29, Number 1 (2001), 92-122.

Dates
First available in Project Euclid: 21 December 2001

Permanent link to this document
https://projecteuclid.org/euclid.aop/1008956323

Digital Object Identifier
doi:10.1214/aop/1008956323

Mathematical Reviews number (MathSciNet)
MR1825143

Zentralblatt MATH identifier
1034.82005

Subjects
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 82B43: Percolation [See also 60K35]

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

Citation

Alexander, Kenneth S. Power-law corrections to exponential decay of connectivities and correlations in lattice models. Ann. Probab. 29 (2001), no. 1, 92--122. doi:10.1214/aop/1008956323. https://projecteuclid.org/euclid.aop/1008956323


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