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Conformal restriction: The chordal case

Authors: Gregory Lawler, Oded Schramm and Wendelin Werner
Journal: J. Amer. Math. Soc. 16 (2003), 917-955
MSC (2000): Primary 60K35, 82B27, 60J69, 30C99
Published electronically: June 2, 2003
MathSciNet review: 1992830
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Abstract: We characterize and describe all random subsets $K$of a given simply connected planar domain (the upper half-plane ${\mathbb H}$, say) which satisfy the ``conformal restriction'' property, i.e., $K$ connects two fixed boundary points ($0$ and $\infty$, say) and the law of $K$ conditioned to remain in a simply connected open subset $H$ of ${\mathbb H}$is identical to that of $\Phi(K)$, where $\Phi$ is a conformal map from ${\mathbb H}$ onto $H$ with $\Phi(0)=0$ and $\Phi(\infty)=\infty$. The construction of this family relies on the stochastic Loewner evolution processes with parameter $\kappa \le 8/3$ and on their distortion under conformal maps. We show in particular that SLE$_{8/3}$ is the only random simple curve satisfying conformal restriction and we relate it to the outer boundaries of planar Brownian motion and SLE$_6$.

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  • 1. L.V. Ahlfors, Conformal Invariants, Topics in Geometric Function Theory, McGraw-Hill, New-York, 1973. MR 50:10211
  • 2. M. Bauer, D. Bernard (2002), SLE$_k$ growth processes and conformal field theories, Phys. Lett. B 543, 135-138.
  • 3. M. Bauer, D. Bernard (2002), Conformal Field Theories of Stochastic Loewner Evolutions, arXiv:hep-th/0210015.
  • 4. V. Beffara (2002), Hausdorff dimensions for SLE$_6$, arXiv:math.PR/0204208.
  • 5. A.A. Belavin, A.M. Polyakov, A.B. Zamolodchikov (1984), Infinite conformal symmetry of critical fluctuations in two dimensions, J. Statist. Phys. 34, 763-774. MR 86e:82019
  • 6. A.A. Belavin, A.M. Polyakov, A.B. Zamolodchikov (1984), Infinite conformal symmetry in two-dimensional quantum field theory. Nuclear Phys. B 241, 333-380. MR 86m:81097
  • 7. R. van den Berg, A. Jarai (2002), The lowest crossing in 2D critical percolation, arXiv: math.PR/0201030.
  • 8. K. Burdzy (1987), Multidimensional Brownian Excursions and Potential Theory, Pitman Research Notes in Mathematics 164, John Wiley & Sons. MR 89d:60146
  • 9. J.L. Cardy (1984), Conformal invariance and surface critical behavior, Nucl. Phys. B240 (FS12), 514-532.
  • 10. J.L. Cardy (1992), Critical percolation in finite geometries, J. Phys. A, 25 L201-L206. MR 92m:82048
  • 11. L. Carleson, N. Makarov (2002), Laplacian path models, J. Anal. Math. 87, 103-150.
  • 12. J. Dubédat (2003), Reflected planar Brownian motions, intertwining relations and crossing probabilities, math.PR/0302250, preprint.
  • 13. J. Dubédat (2003), $SLE(\kappa,\rho)$ martingales and duality, math.PR/0303128, preprint.
  • 14. B. Duplantier (1998), Random walks and quantum gravity in two dimensions, Phys. Rev. Lett. 81, 5489-5492. MR 99j:83034
  • 15. B. Duplantier, K.-H. Kwon (1988), Conformal invariance and intersection of random walks, Phys. Rev. Let. 61, 2514-2517.
  • 16. B. Duplantier, H. Saleur (1986), Exact surface and wedge exponents for polymers in two dimensions, Phys. Rev. Lett. 57, 3179-3182. MR 88e:82022
  • 17. P.L. Duren, Univalent functions, Springer, 1983. MR 85j:30034
  • 18. R. Friedrich, W. Werner (2002), Conformal fields, restriction properties, degenerate representations and SLE, C.R. Ac. Sci. Paris Ser. I Math 335, 947-952.
  • 19. R. Friedrich, W. Werner (2003), Conformal restriction, highest-weight representations and SLE, math-ph/0301018, preprint.
  • 20. T. Kennedy (2002), A faster implementation of the pivot algorithm for self-avoiding walks, J. Stat. Phys. 106, 407-429.
  • 21. T. Kennedy (2002), Monte Carlo tests of SLE predictions for the 2D self-avoiding walk, Phys. Rev. Lett. 88, 130601.
  • 22. T. Kennedy (2002), Conformal Invariance and Stochastic Loewner Evolution Predictions for the 2D Self-Avoiding Walk - Monte Carlo Tests, arXiv:math.PR/0207231.
  • 23. G.F. Lawler, O. Schramm, W. Werner (2001), Values of Brownian intersection exponents I: Half-plane exponents. Acta Mathematica 187, 237-273. MR 2002m:60159a
  • 24. G.F. Lawler, O. Schramm, W. Werner (2001), Values of Brownian intersection exponents II: Plane exponents. Acta Mathematica 187, 275-308. MR 2002m:60159b
  • 25. G.F. Lawler, O. Schramm, W. Werner (2002), Values of Brownian intersection exponents III: Two-sided exponents. Ann. Inst. Henri Poincaré 38, 109-123. MR 2003d:60163
  • 26. G.F. Lawler, O. Schramm, W. Werner (2002), Analyticity of planar Brownian intersection exponents. Acta Mathematica 189, 179-201.
  • 27. G.F. Lawler, O. Schramm, W. Werner (2001), Sharp estimates for Brownian non-intersection probabilities, in In and Out of Equilbrium, V. Sidoravicius, Ed., Prog. Probab., Birkhauser, 113-131. MR 2003d:60162
  • 28. G.F. Lawler, O. Schramm, W. Werner (2002), One-arm exponent for critical 2D percolation, Electronic J. Probab. 7, paper no. 2. MR 2002k:60204
  • 29. G.F. Lawler, O. Schramm, W. Werner (2001), Conformal invariance of planar loop-erased random walks and uniform spanning trees, arXiv:math.PR/0112234, Ann. Prob., to appear.
  • 30. G.F. Lawler, O. Schramm, W. Werner (2002), On the scaling limit of planar self-avoiding walks, math.PR/0204277, in Fractal geometry and application, A jubilee of Benoit Mandelbrot, AMS Proc. Symp. Pure Math., to appear.
  • 31. G.F. Lawler, O. Schramm, W. Werner (2002), Conformal restriction: the radial case, in preparation.
  • 32. G.F. Lawler, W. Werner (2000), Universality for conformally invariant intersection exponents, J. Europ. Math. Soc. 2, 291-328. MR 2002g:60123
  • 33. G.F. Lawler, W. Werner (2003), The Brownian loop soup, preprint.
  • 34. B.B. Mandelbrot, The Fractal Geometry of Nature, Freeman, 1982. MR 84h:00021
  • 35. B. Nienhuis, E.K. Riedel, M. Schick (1980), Magnetic exponents of the two-dimensional $q$-states Potts model in two dimensions, J. Phys A 13, L. 189-192.
  • 36. B. Nienhuis (1984), Critical behavior in two dimensions and charge symmetry of the Coulomb gas, J. Stat. Phys. 34, 731-761.
  • 37. M.P.M. den Nijs (1979), A relation between the temperature exponents of the eight-vertex and the $q$-state Potts model, J. Phys. A 12, 1857-1868.
  • 38. R.P. Pearson (1980), Conjecture for the extended Potts model magnetic eigenvalue, Phys. Rev. B 22, 2579-2580.
  • 39. C. Pommerenke, Univalent functions, Vandenhoeck & Ruprecht, Göttingen, 1975. MR 58:22526
  • 40. C. Pommerenke, Boundary behaviour of conformal maps, Springer-Verlag, Berlin, 1992. MR 95b:30008
  • 41. D. Revuz, M. Yor, Continuous Martingales and Brownian Motion, Springer, 2nd Ed., 1994. MR 95h:60072
  • 42. S. Rohde, O. Schramm (2001), Basic properties of SLE, arXiv:math.PR/0106036, preprint.
  • 43. H. Saleur, B. Duplantier (1987), Exact determination of the percolation hull exponent in two dimensions, Phys. Rev. Lett. 58, 2325. MR 88d:82073
  • 44. O. Schramm (2000), Scaling limits of loop-erased random walks and uniform spanning trees, Israel J. Math. 118, 221-288. MR 2001m:60227
  • 45. O. Schramm (2001), A percolation formula, Electronic Comm. Probab. 6, 115-120. MR 2002h:60227
  • 46. S. Smirnov (2001), Critical percolation in the plane: Conformal invariance, Cardy's formula, scaling limits, C. R. Acad. Sci. Paris Sér. I Math. 333 no. 3, 239-244. MR 2002f:60193
  • 47. S. Smirnov, W. Werner (2001), Critical exponents for two-dimensional percolation, Math. Res. Lett. 8, 729-744.
  • 48. S.R.S. Varadhan, R.J. Williams (1985), Brownian motion in a wedge with oblique reflection. Comm. Pure Appl. Math. 38, 405-443. MR 87c:60066
  • 49. B. Virág (2003), Brownian beads, in preparation.
  • 50. W. Werner (2001), Critical exponents, conformal invariance and planar Brownian motion, in Proceedings of the 4th ECM Barcelona 2000, Prog. Math. 202, Birkhäuser, 87-103. MR 2003f:60181
  • 51. W. Werner (2003), Girsanov's Theorem for SLE $(\kappa,\rho)$ processes, intersection exponents and hiding exponents, math.PR/0302115, preprint.

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Additional Information

Gregory Lawler
Affiliation: Department of Mathematics, 310 Malott Hall, Cornell University, Ithaca, New York 14853-4201

Oded Schramm
Affiliation: Microsoft Corporation, One Microsoft Way, Redmond, Washington 98052

Wendelin Werner
Affiliation: Département de Mathématiques, Bât. 425, Université Paris-Sud, 91405 ORSAY cedex, France

Keywords: Conformal invariance, restriction property, random fractals, SLE
Received by editor(s): October 9, 2002
Published electronically: June 2, 2003
Article copyright: © Copyright 2003 American Mathematical Society

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