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Ornstein-Zernike theory for the Bernoulli bond percolation on \(\mathbb Z^d\). (English) Zbl 1013.60077

Summary: 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 \(p<p_c(d)\).

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
60F15 Strong limit theorems
60K15 Markov renewal processes, semi-Markov processes
82B43 Percolation
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[1] AIZENMAN, M. and BARSKY, D. J. (1987). Sharpness of phase transitions in percolation models. Comm. Math. Phys. 108 489-526. · Zbl 0618.60098
[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. · Zbl 0718.60110
[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. · Zbl 0739.60089
[4] ALEXANDER, K. S. (1997). Approximation of subadditive functions and convergence rates in limiting-shape results. Ann. Probab. 25 30-55. · Zbl 0882.60090
[5] ALEXANDER, K. S. (2001). Power-law corrections to exponential decay of connectivities and correlations in lattice models. Ann. Probab. 29 92-122. · Zbl 1034.82005
[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. · Zbl 0698.60098
[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. · Zbl 0691.60090
[8] CAMPANINO, M., IOFFE, D. and VELENIK, Y. (2001). Ornstein-Zernike theory for finite range Ising models above critical temperature. · Zbl 1032.60093
[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. · Zbl 0815.60024
[11] GRIMMETT, G. R. (1989). Percolation. Springer, New York. · Zbl 0691.60089
[12] IOFFE, D. (1998). Ornstein-Zernike behaviour and analyticity of shapes for self-avoiding walks on Zd. Markov Process. Related Fields 4 323-350. · Zbl 0924.60086
[13] IOFFE, D. and SCHONMANN, R. H. (1998). Dobrushin-Kotecký-Shlosman theorem up to the critical temperature I. Comm. Math. Phys. 199 117-167. · Zbl 0929.60076
[14] MADRAS, N. and SLADE, G. (1993). The Self-Avoiding Random Walk. Birkhäuser, Boston. · Zbl 0780.60103
[15] MENSHIKOV, M. V. (1986). Coincidence of critical points in percolation problems. Soviet Math. Doklady 24 856-859. · Zbl 0615.60096
[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. · Zbl 0798.52001
[18] THOMPSON, C. J. (1988). Classical Equilibrium Statistical Mechanics. Calderon Press, Oxford.
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