The Annals of Probability

Ornstein-Zernike theory for the Bernoulli bond percolation on $\mathbb{Z}^d$

Abstract

We derive a precise Ornstein–Zernike asymptotic formula for the decay of the two-point function $\mathbb{P}_p (0 \leftrightarrow x)$ of the Bernoulli bond percolation on the integer lattice $\mathbb{Z}^d$ in any dimension $d \geq 2$, in any direction $x$ and for any subcritical value of $p < p_c (d)$.

Article information

Source
Ann. Probab., Volume 30, Number 2 (2002), 652-682.

Dates
First available in Project Euclid: 7 June 2002

https://projecteuclid.org/euclid.aop/1023481005

Digital Object Identifier
doi:10.1214/aop/1023481005

Mathematical Reviews number (MathSciNet)
MR1905854

Zentralblatt MATH identifier
1013.60077

Citation

Campanino, Massimo; Ioffe, Dmitry. Ornstein-Zernike theory for the Bernoulli bond percolation on $\mathbb{Z}^d$. Ann. Probab. 30 (2002), no. 2, 652--682. doi:10.1214/aop/1023481005. https://projecteuclid.org/euclid.aop/1023481005

References

• [1] AIZENMAN, M. and BARSKY, D. J. (1987). Sharpness of phase transitions in percolation models. Comm. Math. Phys. 108 489-526.
• [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 of the Wulff minimum 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 of subadditive functions and convergence rates in limiting-shape results. Ann. Probab. 25 30-55.
• [5] ALEXANDER, K. S. (2001). Power-law corrections to exponential decay of connectivities and correlations in lattice models. Ann. Probab. 29 92-122.
• [6] ALEXANDER, K. S., CHAYES, J. T. and CHAYES, L. (1990). The Wulff construction and asymptotics of the finite cluster distribution for two-dimensional Bernoulli percolation. Comm. Math. Phys. 131 1-50.
• [7] CAMPANINO, M., CHAYES, J. T. and CHAYES, L. (1991). Gaussian fluctuations in the subcritical regime of percolation. Probab. Theory Related Fields 88 269-341.
• [8] CAMPANINO, M., IOFFE, D. and VELENIK, Y. (2001). Ornstein-Zernike theory for finite range Ising models above critical temperature. Preprint.
• [9] CHAYES, J. T. and CHAYES, L. (1986). Ornstein-Zernike behavior for self-avoiding walks at all noncritical temperatures. Comm. Math. Phys. 105 221-238.
• [10] DOBRUSHIN, R. L. and SHLOSMAN, S. (1994). Large and moderate deviations in the Ising model. Adv. Soviet Math. 20 91-219.
• [11] GRIMMETT, G. R. (1989). Percolation. Springer, New York.
• [12] IOFFE, D. (1998). Ornstein-Zernike behaviour and analyticity of shapes for self-avoiding walks on Zd. Markov Process. Related Fields 4 323-350.
• [13] IOFFE, D. and SCHONMANN, R. H. (1998). Dobrushin-Kotecký­Shlosman theorem up to the critical temperature I. Comm. Math. Phys. 199 117-167.
• [14] MADRAS, N. and SLADE, G. (1993). The Self-Avoiding Random Walk. Birkhäuser, Boston.
• [15] MENSHIKOV, M. V. (1986). Coincidence of critical points in percolation problems. Soviet Math. Doklady 24 856-859.
• [16] ORNSTEIN, L. S. and ZERNIKE, F. (1915). Proc. Acad. Sci. (Amst.) 17 793-806.
• [17] SCHNEIDER, R. (1993). Convex bodies: the Brunn-Minkowski theory. In Encyclopedia of Mathematics and Its Applications 43. Addison-Wesley, Reading, MA.
• [18] THOMPSON, C. J. (1988). Classical Equilibrium Statistical Mechanics. Calderon Press, Oxford.