The Annals of Probability

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

Kenneth S. Alexander

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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

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

First available in Project Euclid: 21 December 2001

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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 82B43: Percolation [See also 60K35]

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


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.

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  • [1] Aizenman, M., Chayes, J. T., Chayes, L. and Newman, C. M. (1988). Discontinuity ofthe magnetization in the 1/ x y 2 Ising and Potts models. J. Statist. Phys. 50 1-40.
  • [2] Alexander, K. S. (1990). Lower bounds on the connectivity function in all directions for Bernoulli percolation in two and three dimensions. Ann. Probab. 18 1547-1562.
  • [3] Alexander, K. S. (1992). Stability ofthe Wulffminimum and fluctuations in shape for large finite clusters in two-dimensional percolation. Probab. Theory Related Fields 91 507- 532.
  • [4] Alexander, K. S. (1997). Approximation ofsubadditive functions and convergence rates in limiting shape results. Ann. Probab. 25 30-55.
  • [5] Alexander, K. S. (1997). The asymmetric random cluster model and comparison ofIsing and Potts models. Probab. Theory Related Fields. To appear.
  • [6] Alexander, K. S (1998). On weak mixing in lattice models. Probab. Theory Related Fields 110 441-471.
  • [7] Alexander, K. S., Chayes, J. T. and Chayes, L. (1990). The Wulff construction and asymptotics ofthe finite cluster distribution for two dimensional Bernoulli percolation. Comm. Math. Phys. 131 1-50.
  • [8] van den Berg, J. and Kesten, H. (1985). Inequalities with applications to percolation and reliability. J. Appl. Probab. 22 556-569.
  • [9] Bricmont, J. and Fr ¨olich, J. (1985). Statistical mechanical methods in particle structure analysis oflattice field theories. II. Scalar and surface models. Comm. Math. Phys. 98 553-578.
  • [10] Campanino, M., Chayes, J. T. and Chayes, L. (1991). Gaussian fluctuations ofconnectivities in the subcritical regime ofpercolation. Probab. Theory Related Fields 88 269-341.
  • [11] Campanino, M. and Ioffe, D. (1999). Ornstein-Zernike theory for the Bernoulli bond percolation on d. Unpublished manuscript.
  • [12] Chayes, J. T. and Chayes, L. (1986). OrnsteinZernike behavior for self-avoiding walks at all noncritical temperatures. Comm. Math. Phys. 105 221-238.
  • [13] Fortuin, C. M. and Kasteleyn, P. W. (1972). On the random cluster model. I. Introduction and relation to other models. Physica 57 536-564.
  • [14] Grimmett, G. R. (1995). The stochastic random-cluster process and uniqueness ofrandomcluster measures. Ann. Probab. 23 1461-1510.
  • [15] Grimmett, G. R. (1997). Percolation and disordered systems. Lectures on Probability Theory and Statistics. Lecture Notes in Math. 1665 153-300. Springer, New York.
  • [16] Harris, T. (1960). A lower bound for the critical probability in a certain percolation process. Proc. Cambridge Philos. Soc. 56 13-20.
  • [17] Ioffe, D. (1998). Ornstein-Zernike behaviour and analyticity of shapes for self-avoiding walks on d. Unpublished manuscript. Markov Processes and Related Fields 4 323-350.
  • [18] Laanait, L., Messager, A. and Ruiz, J. (1986). Phase coexistence and surface tensions for the Potts model. Comm. Math. Phys. 105 527-545.
  • [19] McCoy, B. M. and Wu, T. T. (1973). The Two-Dimensional Ising Model. Harvard Univ. Press.
  • [20] Ornstein, L. S. and Zernike, F. (1914). Accidental deviations ofdensity and opalescence at the critical point ofa single substance. Proc. Acad. Sci. Amst. 17 793-806.
  • [21] Pfister, C.-E. and Velenik, Y. (1997). Large deviations and continuum limit in the 2D Ising model. Probab. Theory Related Fields 109 435-506.