Probability Theory and Related Fields

, Volume 134, Issue 3, pp 453–488 | Cite as

Excursion decompositions for SLE and Watts' crossing formula

Article

Abstract

It is known that Schramm-Loewner Evolutions (SLEs) have a.s. frontier points if κ>4 and a.s. cutpoints if 4<κ<8. If κ>4, an appropriate version of SLE(κ) has a renewal property: it starts afresh after visiting its frontier. Thus one can give an excursion decomposition for this particular SLE(κ) “away from its frontier”. For 4<κ<8, there is a two-sided analogue of this situation: a particular version of SLE(κ) has a renewal property w.r.t its cutpoints; one studies excursion decompositions of this SLE “away from its cutpoints”. For κ=6, this overlaps Virág's results on “Brownian beads”. As a by-product of this construction, one proves Watts' formula, which describes the probability of a double crossing in a rectangle for critical plane percolation.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Beffara, V.: Hausdorff dimensions for SLE6. Ann. Probab. 32 (3B), 2606–2629 (2004)MATHCrossRefGoogle Scholar
  2. 2.
    Belavin, A.A., Polyakov, A.M., Zamolodchikov, A.B.: Infinite conformal symmetry in two-dimensional quantum field theory. Nuclear Phys. B, 241 (2), 333–380 (1984)CrossRefGoogle Scholar
  3. 3.
    Bertoin, J.: Complements on the Hilbert transform and the fractional derivative of Brownian local times. J. Math. Kyoto Univ. 30 (4), 651–670 (1990)Google Scholar
  4. 4.
    Bertoin, J.: Excursions of a BES0(d) and its drift term (0<d<1). Probab. Theory Relat. Fields, 84 (2), 231–250 (1990)CrossRefGoogle Scholar
  5. 5.
    Cardy, J.L.: Critical percolation in finite geometries. J. Phys. A 25 (4), L201–L206 (1992)Google Scholar
  6. 6.
    Cardy, J.L.: Conformal invariance and percolation. preprint, arXiv:math-ph/0103018, 2001Google Scholar
  7. 7.
    Cardy, J.L.: Crossing formulae for critical percolation in an annulus. J. Phys. A 35 (41), L565–L572, (2002)Google Scholar
  8. 8.
    Carmona, P., Petit, F., Yor, M.: Beta-gamma random variables and intertwining relations between certain Markov processes. Rev. Mat. Iberoamericana 14 (2), 311–367 (1998)Google Scholar
  9. 9.
    Dubédat, J.: Reflected brownian motions, intertwining relations and crossing probabilities. Ann. Inst. H. Poincaré Probab. Statist. to appear, 2003Google Scholar
  10. 10.
    Dubédat, J.: SLE(κ,ρ) martingales and duality. Ann. Probab. to appear, 2003Google Scholar
  11. 11.
    Dubédat, J.: Critical percolation in annuli and SLE6. Comm. Math. Phys. 245 (3), 627–637 (2004)CrossRefGoogle Scholar
  12. 12.
    Grimmett, G.: Percolation, volume 321 of Grundlehren der Mathematischen Wissenschaften. Springer-Verlag, Berlin, second edition, 1999Google Scholar
  13. 13.
    Itô, K., McKean, H.P. Jr.: Diffusion processes and their sample paths. Springer-Verlag, Berlin, 1974. Second printing, corrected, Die Grundlehren der mathematischen Wissenschaften, Band 125Google Scholar
  14. 14.
    Kleban, P., Zagier, D.: Crossing probabilities and modular forms. J. Statist. Phys. 113 (3–4), 431–454 (2003)Google Scholar
  15. 15.
    Langlands, R., Pouliot, P., Saint-Aubin, Y.: Conformal invariance in two-dimensional percolation. Bull. Amer. Math. Soc. (N.S.) 30 (1), 1–61 (1994)Google Scholar
  16. 16.
    Lawler, G, Schramm, O., Werner, W.: Conformal restriction: the chordal case. J. Amer. Math. Soc. 16 (4), 917–955 (electronic), (2003)CrossRefGoogle Scholar
  17. 17.
    Lawler, G.F., Schramm, O., Werner, W.: Values of Brownian intersection exponents. III. Two-sided exponents. Ann. Inst. H. Poincaré Probab. Statist. 38 (1), 109–123 (2002)CrossRefGoogle Scholar
  18. 18.
    Lawler, G.F., Schramm, O., Werner, W.: Conformal invariance of planar loop-erased random walks and uniform spanning trees. Ann. Probab. 32 (1B), 939–995 (2004)MATHCrossRefGoogle Scholar
  19. 19.
    Maier, R.S.: On crossing event formulas in critical two-dimensional percolation. J. Statist. Phys. 111 (5–6), 1027–1048 (2003)Google Scholar
  20. 20.
    Pinson, H.T.: Critical percolation on the torus. J. Statist. Phys. 75 (5–6), 1167–1177 (1994)Google Scholar
  21. 21.
    Pitman, J., Yor, M.: A decomposition of Bessel bridges. Z. Wahrsch. Verw. Gebiete 59 (4), 425–457 (1982)CrossRefGoogle Scholar
  22. 22.
    Revuz, D., Yor, M.: Continuous martingales and Brownian motion. Volume 293 of Grundlehren der Mathematischen Wissenschaften. Springer-Verlag, Berlin, third edition, 1999Google Scholar
  23. 23.
    Rogers, L.C.G., Williams, D.: Diffusions, Markov processes, and martingales. Vol. 1. John Wiley & Sons Ltd., Chichester, second edition, 1994Google Scholar
  24. 24.
    Rohde, S., Schramm, O.: Basic Properties of SLE. Ann. Math. to appear, 2001Google Scholar
  25. 25.
    Schramm, O.: Scaling limits of loop-erased random walks and uniform spanning trees. Israel J. Math. 118, 221–288 (2000)MATHGoogle Scholar
  26. 26.
    Smirnov, S.: Critical percolation in the plane. I. Conformal Invariance and Cardy's formula II. Continuum scaling limit. in preparation, 2001Google Scholar
  27. 27.
    Taylor, S.J., Wendel, J.G.: The exact Hausdorff measure of the zero set of a stable process. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 6, 170–180 (1966)MATHCrossRefGoogle Scholar
  28. 28.
    Virág, B.: Brownian beads. Probab. Theory Relat. Fields 127 (3), 367–387 (2003)CrossRefGoogle Scholar
  29. 29.
    Watts, G.M.T.: A crossing probability for critical percolation in two dimensions. J. Phys. A 29 (14), L363–L368 (1996)Google Scholar
  30. 30.
    Werner, W.: Girsanov's transformation for SLE(κ,ρ) processes, intersection exponents and hiding exponents. Ann. Fac. Sci. Toulouse Math. (6) 13 (1), 121–147 (2004)Google Scholar
  31. 31.
    Werner, W.: Lectures on random planar curves and Schramm-Loewner evolution. In: Lecture Notes of the 2002 St-Flour summer school, number 1840 in Lecture Notes in Math., Springer-Verlag, 2004, pp. 107–195Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  1. 1.Courant InstituteNew YorkUSA

Personalised recommendations