Electronic Communications in Probability

A Percolation Formula

Oded Schramm

Full-text: Open access

Abstract

Let $A$ be an arc on the boundary of the unit disk $U$. We prove an asymptotic formula for the probability that there is a percolation cluster $K$ for critical site percolation on the triangular grid in $U$ which intersects $A$ and such that $0$ is surrounded by the union of $K$ and $A$.

Article information

Source
Electron. Commun. Probab., Volume 6 (2001), paper no. 12, 115-120.

Dates
Accepted: 24 October 2001
First available in Project Euclid: 19 April 2016

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1461097557

Digital Object Identifier
doi:10.1214/ECP.v6-1041

Mathematical Reviews number (MathSciNet)
MR1871700

Zentralblatt MATH identifier
1008.60100

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 30C35: General theory of conformal mappings

Keywords
SLE Cardy conformal invariance

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Schramm, Oded. A Percolation Formula. Electron. Commun. Probab. 6 (2001), paper no. 12, 115--120. doi:10.1214/ECP.v6-1041. https://projecteuclid.org/euclid.ecp/1461097557


Export citation

References

  • Cardy, John L. Critical percolation in finite geometries. J. Phys. A 25 (1992), no. 4, L201–L206.
  • John L. Cardy. Conformal Invariance and Percolation.
  • Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger, and Francesco G. Tricomi. Higher transcendental functions. Vol. I. McGraw-Hill Book Company, Inc., New York-Toronto-London, 1953. Based, in part, on notes left by Harry Bateman.
  • Grimmett, Geoffrey. Percolation. Springer-Verlag, New York, 1989. xii+296 pp. ISBN: 0-387-96843-1.
  • Kesten, Harry. Percolation theory for mathematicians. Progress in Probability and Statistics, 2. Birkhäuser, Boston, Mass., 1982. iv+423 pp. ISBN: 3-7643-3107-0.
  • Langlands, Robert; Pouliot, Philippe; Saint-Aubin, Yvan. Conformal invariance in two-dimensional percolation. Bull. Amer. Math. Soc. (N.S.) 30 (1994), no. 1, 1–61.
  • Lawler, Gregory F.; Schramm, Oded; Werner, Wendelin. Values of Brownian intersection exponents. I. Half-plane exponents. Acta Math. 187 (2001), no. 2, 237–273.
  • Rohde, Steffen; Schramm, Oded. Basic properties of SLE. Ann. of Math. (2) 161 (2005), no. 2, 883–924.
  • Schramm, Oded. Scaling limits of loop-erased random walks and uniform spanning trees. Israel J. Math. 118 (2000), 221–288.
  • Stanislav Smirnov. Critical percolation in the plane. I. Conformal invariance and Cardy's formula. II. Continuum scaling limit. Preprint.
  • Stanislav Smirnov. In preparation.
  • Watts, G. M. T. A crossing probability for critical percolation in two dimensions. J. Phys. A 29 (1996), no. 14, L363–L368.