## The Annals of Probability

### On the scaling of the chemical distance in long-range percolation models

Marek Biskup

#### Abstract

We consider the (unoriented) long-range percolation on ℤd in dimensions d≥1, where distinct sites x,y∈ℤd get connected with probability pxy∈[0,1]. Assuming pxy=|xy|s+o(1) as |xy|→∞, where s>0 and |⋅| is a norm distance on ℤd, and supposing that the resulting random graph contains an infinite connected component C, we let D(x,y) be the graph distance between x and y measured on C. Our main result is that, for s∈(d,2d), D(x,y)=(log|xy|)Δ+o(1),  x,yC, |xy|→∞, where Δ−1 is the binary logarithm of 2d/s and o(1) is a quantity tending to zero in probability as |xy|→∞. Besides its interest for general percolation theory, this result sheds some light on a question that has recently surfaced in the context of “small-world” phenomena. As part of the proof we also establish tight bounds on the probability that the largest connected component in a finite box contains a positive fraction of all sites in the box.

#### Article information

Source
Ann. Probab. Volume 32, Number 4 (2004), 2938-2977.

Dates
First available in Project Euclid: 8 February 2005

http://projecteuclid.org/euclid.aop/1107883343

Digital Object Identifier
doi:10.1214/009117904000000577

Mathematical Reviews number (MathSciNet)
MR2094435

Zentralblatt MATH identifier
1072.60084

#### Citation

Biskup, Marek. On the scaling of the chemical distance in long-range percolation models. Ann. Probab. 32 (2004), no. 4, 2938--2977. doi:10.1214/009117904000000577. http://projecteuclid.org/euclid.aop/1107883343.

#### References

• Aizenman, M. and Barsky, D. J. (1987). Sharpness of the phase transition in percolation models. Comm. Math. Phys. 108 489--526.
• Aizenman, M. and Newman, C. (1986). Discontinuity of the percolation density in one-dimensional $1/|x-y|^2$ percolation models. Comm. Math. Phys. 107 611--647.
• Alexander, K., 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--51.
• Antal, P. (1994). Trapping problems for the simple random walk. Dissertation ETH No. 10759.
• Antal, P. and Pisztora, A. (1996). On the chemical distance for supercritical Bernoulli percolation. Ann. Probab. 24 1036--1048.
• Benjamini, I. and Berger, N. (2001). The diameter of long-range percolation clusters on finite cycles. Random Structures Algorithms 19 102--111.
• Benjamini, I., Kesten, H., Peres, Y. and Schramm, O. (2004). The geometry of the uniform spanning forests: Transitions in dimensions 4, 8, 12, $\ldots\,$. Ann. Math. To appear.
• Berger, N. (2002). Transience, recurrence and critical behavior for long-range percolation. Comm. Math. Phys. 226 531--558.
• Bollobás, B. (2001). Random Graphs. Cambridge Univ. Press.
• Burton, R. M. and Keane, M. (1989). Density and uniqueness in percolation. Comm. Math. Phys. 121 501--505.
• Campanino, M., Chayes, J. T. and Chayes, L. (1991). Gaussian fluctuations in the subcritical regime of percolation. Probab. Theory Related Fields 88 269--341.
• Campanino, M. and Ioffe, D. (2002). Ornstein--Zernike theory for the Bernoulli bond percolation on $Z^d$. Ann. Probab. 30 652--682.
• Cerf, R. (2000). Large deviations for three dimensional supercritical percolation. Astérisque 267 vi+177.
• Coppersmith, D., Gamarnik, D. and Sviridenko, M. (2002). The diameter of a long-range percolation graph. Random Structures Algorithms 21 1--13.
• Gandolfi, A., Keane, M. S. and Newman, C. M. (1992). Uniqueness of the infinite component in a random graph with applications to percolation and spin glasses. Probab. Theory Related Fields 92 511--527.
• Grimmett, G. R. and Marstrand, J. M. (1990). The supercritical phase of percolation is well behaved. Proc. Roy. Soc. London Ser. A 430 439--457.
• Hara, T. and Slade, G. (2000). The scaling limit of the incipient infinite cluster in high-dimensional percolation. I. Critical exponents. J. Statist. Phys. 99 1075--1168.
• Hara, T. and Slade, G. (2000). The scaling limit of the incipient infinite cluster in high-dimensional percolation. II. Integrated super-Brownian excursion. J. Math. Phys. 41 1244--1293.
• Imbrie, J. and Newman, C. (1988). An intermediate phase with slow decay of correlations in one-dimensional $1/\vert x-y\vert\sp2$ percolation, Ising and Potts models. Comm. Math. Phys. 118 303--336.
• Menshikov, M. V. (1986). Coincidence of critical points in percolation problems. Dokl. Akad. Nauk SSSR 288 1308--1311. (In Russian.)
• Milgram, S. (1967). The small-world problem. Psychology Today 1 61--67.
• Newman, C. M. and Schulman, L. S. (1986). One-dimensional $1/\vert j-i\vert\sp s$ percolation models: The existence of a transition for $s\leq2$. Comm. Math. Phys. 104 547--571.
• Schulman, L. S. (1983). Long-range percolation in one dimension. J. Phys. A 16 L639--L641.
• Smirnov, S. (2001). Critical percolation in the plane. I. Conformal invariance and Cardy's formula. II. Continuum scaling limit. C. R. Acad. Sci. Paris Sér. I Math. 333 239--244.
• Smirnov, S. and Werner, W. (2001). Critical exponents for two-dimensional percolation. Math. Res. Lett. 8 729--744.
• Watts, D. J. and Strogatz, S. H. (1998). Collective dynamics of small-world'' networks. Nature 303 440--442.