Abstract
Building on the identification of the scaling limit of the critical percolation exploration process as a Schramm-Loewner Evolution, we derive a PDE characterization for the crossing probability of an annulus.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Ahlfors, L.: Complex Analysis. 3rd edition, New York: McGraw-Hill, 1979
Cardy, J.L.: Critical percolation in finite geometries. J. Phys. A 25, L201–L206 (1992)
Cardy, J.L.: Conformal invariance and percolation. Preprint, arXiv:math-ph/0103018, 2001
Cardy, J.L.: Crossing Formulae for Critical Percolation in an Annulus. Preprint, arXiv:math-ph/0208019v4, 2002
Chandrasekharan, K.: Elliptic functions. Grundlehren der mathematischen Wissenschaften 281, Berlin-Heidelberg-New York: Springer-Verlag, 1984
Dubédat, J.: In preparation
Friedrich, R., Kalkkinen, J.: In preparation
Grimmett, G.R.: Percolation and disordered systems. In Lectures on Probability Theory and Statistics, Ecole d’été de probabilités de Saint-Flour XXVI, Lecture Notes in Mathematics 1665, Berlin-Heidelberg-New York: Springer-Verlag, 1997
Lawler, G.: An Introduction to the Stochastic Loewner Evolution. Preprint, 2001
Lawler, G., Schramm, O., Werner, W.: Values of Brownian intersection exponents I: Half-plane exponents. Acta Math. 187, 237–273 (2001)
Lawler, G., Schramm, O., Werner, W.: Values of Brownian intersection exponents II: Plane exponents. Acta Math. 187, 275–308 (2001)
Lawler, G., Schramm, O., Werner, W.: Values of Brownian intersection exponents III: Two-sided exponents. Ann. Inst. H. Poincaré Probab. Statist. 38(1), 109–123 (2002)
Lawler, G., Schramm, O., Werner, W.: Conformal Invariance of planar loop-erased random walks and uniform spanning trees, arXiv:math.PR/0112234, Ann. Probab., to appear
Lawler, G., Schramm, O., Werner, W.: On the scaling limit of planar self-avoiding walk, math.PR/0204277, 2002
Lawler, G., Schramm, O., Werner, W.: Conformal restriction. The chordal case. arXiv:math.PR/0209343, J. Amer. Math. Soc., to appear
Lawler, G., Schramm, O., Werner, W.: One-arm exponent for critical 2D percolation. Electr. J. Probab. 7(2), 2002
Pinson, H.: Critical percolation on the torus. J. Statist. Phys. 75(5-6), 1167–1177 (1994)
Revuz, D., Yor, M.: Continuous martingales and Brownian motion. 2nd edition, Grundlehren der mathematischen Wissenschaften 293, Berlin-Heidelberg-New-York: Springer- Verlag, 1994
Rohde, S., Schramm, O.: Basic Properties of SLE. arXiv:math.PR/0106036, 2001
Schramm, O.: Scaling limits of loop-erased random walks and uniform spanning trees. Israel J. Math. 118, 221–288 2000
Schramm, O.: A percolation formula. Electr. Comm. Probab. 6, 115–120
Smirnov, S.: Critical percolation in the plane. I. Conformal Invariance and Cardy’s formula II. Continuum scaling limit. In preparation, 2001
Smirnov, S., Werner, W.: Critical exponents for two-dimensional percolation. Math. Res. Lett. 8, 729–744 (2001)
Villat, H.: Le problème de Dirichlet dans une aire annulaire. Rend. circ. mat. Palermo, 134–175 (1912)
Watts, G.M.T.: A crossing probability for critical percolation in two dimensions. J. Phys. A: Math. Gen. 29, 363–368 1996
Werner, W.: Random planar curves and Schramm-Loewner evolution. Lecture Notes of the 2002 St-Floor summer school, Springer, to appear
Zhan, D.: In preparation
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by A. Kupiainen
Rights and permissions
About this article
Cite this article
Dubédat, J. Critical Percolation in Annuli and SLE 6 . Commun. Math. Phys. 245, 627–637 (2004). https://doi.org/10.1007/s00220-003-1029-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-003-1029-8