Electronic Journal of Probability

On the chemical distance in critical percolation

Abstract

We consider two-dimensional critical bond percolation. Conditioned on the existence of an open circuit in an annulus, we show that the ratio of the expected size of the shortest open circuit to the expected size of the innermost circuit tends to zero as the side length of the annulus tends to infinity, the aspect ratio remaining fixed. The same proof yields a similar result for the lowest open crossing of a rectangle. In this last case, we answer a question of Kesten and Zhang by showing in addition that the ratio of the length of the shortest crossing to the length of the lowest tends to zero in probability. This suggests that the chemical distance in critical percolation is given by an exponent strictly smaller than that of the lowest path.

Article information

Source
Electron. J. Probab., Volume 22 (2017), paper no. 75, 43 pp.

Dates
Accepted: 3 August 2017
First available in Project Euclid: 14 September 2017

https://projecteuclid.org/euclid.ejp/1505354464

Digital Object Identifier
doi:10.1214/17-EJP88

Mathematical Reviews number (MathSciNet)
MR3698744

Zentralblatt MATH identifier
06797885

Citation

Damron, Michael; Hanson, Jack; Sosoe, Philippe. On the chemical distance in critical percolation. Electron. J. Probab. 22 (2017), paper no. 75, 43 pp. doi:10.1214/17-EJP88. https://projecteuclid.org/euclid.ejp/1505354464

References

• [1] Aizenman, M., Burchard, A.: Hölder regularity and dimension bounds on random curves. Duke Math. J. 99, (1999), 419 – 453.
• [2] Antal, P., Pisztora, A: On the chemical distance for supercritical Bernoulli percolation. Ann. Probab. 24, (1996), 1036–1048.
• [3] Beffara, V.: The dimension of SLE curves. Ann. Probab. 4, (2008), 1421–1452.
• [4] van den Berg, J., Fiebig, U.: On a combinatorial conjecture concerning disjoint occurrences of events. Ann. Probab. 15, (1987), 354–374.
• [5] Damron, M., Hanson, J., Sosoe, P.: On the Chemical Distance in Critical Percolation. arXiv:1506.03461.
• [6] Damron, M, Hanson, J., Sosoe, P.: On the Chemical Distance in Critical Percolation II. arXiv:1601.03464.
• [7] Damron, M., Sapozhnikov, A.: Outlets of 2D invasion percolation and multiple-armed incipient infinite clusters. Probab. Theory Related Fields 150, (2011), 257–294.
• [8] Grassberger, P.: Pair connectedness and the shortest-path scaling in critical percolation. J. Phys. A 32, (1999), 6233–6238. 32, 1999.
• [9] Grimmett, G. R., Marstrand, J. M.: The supercritical phase of percolation is well-behaved. Proc. Roy. Soc. London Ser. A 430, (1990), 439 – 457.
• [10] Herrmann, H.-J., Stanley, H.-E.: The fractal dimension of the minimum path in two- and three-dimensional percolation. J. Phys. A 21, (1988), L829–L833.
• [11] Heydenreich, M., van der Hofstad, R., Hulshof, T.: Random walk on the high-dimensional IIC. Comm. Math. Phys. 329, (2014), 57–115.
• [12] van der Hofstad, R., Sapozhnikov, A.: Cycle structure of percolation on high-dimensional tori. Ann. Inst. Henri Poincaré Probab. Stat. 50, (2014), 999–1027.
• [13] Kesten, H.: Percolation Theory for Mathematicians. Progress in Probability and Statistics, 2. Birkhauser-Boston, Mass., 1982. iv+423 pp. ISBN: 3-7643-3107-0.
• [14] Kesten, H.: The critical probability of bond percolation on the square lattice equals $1\over 2$. Comm. Math. Phys. 74, (1980), 41–59.
• [15] Kesten, H.: Scaling relations for $2$D-percolation. Comm. Math. Phys. 109, (1987), 109–156.
• [16] Kesten, H., Zhang, Y.: The tortuosity of occupied crossings of a box in critical percolation. J. Statist. Phys. 70, (1993), 599–611.
• [17] Kesten, H., Sidoravicius, V., Zhang, Y.: Almost all words are seen in critical site percolation on the triangular lattice. Electron. J. Probab. 3, (1998), 1–75.
• [18] Kiss, D.; Manolescu, I.; Sidoravicius, V.: Planar lattices do not recover from forest fires. Ann. Probab. 43, (2015), 3216–3238.
• [19] Kozma, G., Nachmias, A.: The Alexander-Orbach conjecture holds in high dimensions. Invent. Math. 178, (2009), 635–654.
• [20] Kozma, G., Nachmias, A.: Arm exponents in high dimensional percolation. J. Amer. Math. Soc. 24, (2011), 375–409.
• [21] Morrow, G. J., Zhang, Y.: The sizes of the pioneering, lowest crossing and pivotal sites in critical percolation on the triangular lattice. Ann. Appl. Probab. 15, (2005), 1832–1886.
• [22] Nguyen, B. G.: Typical cluster size for two-dimensional percolation processes. J. Statist. Phys. 50, (1988), 715–726.
• [23] Nolin, P.: Near-critical percolation in two dimensions. Electron. J. Probab. 13, (2008), 1562–1623.
• [24] Pisztora, A.: Scaling inequalities for shortest paths in regular and invasion percolation. Carnegie-Mellon CNA preprint, available at http://www.math.cmu.edu/CNA/Publications/publications2000/001abs/00-CNA-001.pdf
• [25] Posé, N., Schrenk, K. J., Araujo, N. A. M., Herrmann, H. J.: Shortest path and Schramm-Loewner Evolution. Scientific Reports 4, (2014).
• [26] Reimer, D.: Proof of the Van den Berg-Kesten conjecture. Combin. Probab. Comput. 9, (2000), 27–32.
• [27] Russo, L.: A note on percolation. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 43, (1978), 39–48.
• [28] Schramm, O.: Conformally invariant scaling limits: an overview and a collection of problems. Selected works of Oded Schramm. Volume 1, 2, 1161–1191, Sel. Works Probab. Stat., Springer, New York, 2011.
• [29] Schramm, O., Werner, W.: Critical exponents for two-dimensional percolation. Math. Res. Lett. 8, (2001), 729–744.
• [30] Seymour, P.D., Welsh, D.J.A.: Percolation probabilities on the square lattice. Advances in graph theory (Cambridge Combinatorial Conf., Trinity College, Cambridge, 1977). Ann. Discrete Math. 3, (1978), 227–245.
• [31] van den Berg, J., Járai, A.A.: The lowest crossing in critical percolation. Ann. Probab. 31, (2003), 1241–1253.
• [32] Zhou, Z., Yang, J., Deng, Y., and Ziff, R. M.: Shortest-path fractal dimension for percolation in two and three dimensions. Phys. Rev. E 86, (2012).