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Values of Brownian intersection exponents, II: Plane exponents

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References

  1. Ahlfors, L. V.,Conformal Invariants: Topics in Geometric Function Theory. McGraw-Hill, New York, 1973.

    Google Scholar 

  2. Aizenman, M., Duplantier, B. &Aharony, A., Path crossing exponents and the external perimeter in 2D percolation.Phys. Rev. Lett., 83 (1999), 1359–1362.

    Article  Google Scholar 

  3. Azencott, R., Behaviour of diffusion semi-groups at infinity.Bull. Soc. Math. France, 102 (1974), 193–240.

    MATH  MathSciNet  Google Scholar 

  4. Bishop, C. J., Jones, P. W., Pemantle, R. &Peres, Y., The dimension of the Brownian frontier is greater than 1.J. Funct. Anal., 143 (1997), 309–336.

    Article  MathSciNet  Google Scholar 

  5. Burdzy, K. &Lawler, G. F., Non-intersection exponents for Brownian paths. Part I: Existence and an invariance principle.Probab. Theory Related Fields, 84 (1990), 393–410.

    Article  MathSciNet  Google Scholar 

  6. —, Non-intersection exponents for Brownian paths. Part II: Estimates and applications to a random fractal.Ann. Probab., 18 (1990), 981–1009.

    MathSciNet  Google Scholar 

  7. Cardy, J. L., Conformal invariance and surface critical behavior.Nuclear Phys. B, 240 (1984), 514–532.

    Article  Google Scholar 

  8. —, Critical percolation in finite geometries.J. Phys. A, 25 (1992), L201-L206.

    Article  MATH  MathSciNet  Google Scholar 

  9. —, The number of incipient spanning clusters in two-dimensional percolation.J. Phys. A, 31 (1998), L105.

    Article  MATH  Google Scholar 

  10. Cranston, M. &Mountford, T., An extension of a result of Burdzy and Lawler.Probab. Theory Related Fields, 89 (1991), 487–502.

    Article  MathSciNet  Google Scholar 

  11. Duplantier, B., Random walks and quantum gravity in two dimensions.Phys. Rev. Lett., 81 (1998), 5489–5492.

    Article  MATH  MathSciNet  Google Scholar 

  12. —, Two-dimensional copolymers and exact conformal multifractality.Phys. Rev. Lett., 82 (1999), 880–883.

    Article  MathSciNet  Google Scholar 

  13. —, Harmonic measure exponents for two-dimensional percolation.Phys. Rev. Lett., 82 (1999), 3940–3943.

    Article  MATH  MathSciNet  Google Scholar 

  14. Duplantier, B. &Kwon, K.-H., Conformal invariance and intersection of random walks.Phys. Rev. Lett., 61 (1988), 2514–2517.

    Article  Google Scholar 

  15. Duplantier, B. &Saleur, H., Exact determination of the percolation hull exponent in two dimensions.Phys. Rev. Lett., 58 (1987), 2325–2328.

    Article  MathSciNet  Google Scholar 

  16. Kenyon, R., Conformal invariance of domino tiling.Ann. Probab., 28 (2000), 759–795.

    Article  MATH  MathSciNet  Google Scholar 

  17. —, Long-range properties of spanning trees. Probabilistic techniques in equilibrium and nonequilibrium statistical physics.J. Math. Phys., 41 (2000), 1338–1363.

    Article  MATH  MathSciNet  Google Scholar 

  18. —, The asymptotic determinant of the discrete Laplacian.Acta Math., 185 (2000), 239–286.

    MATH  MathSciNet  Google Scholar 

  19. Lawler, G. F.,Intersections of Random Walks. Birkhäuser Boston, Boston, MA, 1991.

    Google Scholar 

  20. —, Hausdorff dimension of cut points for Brownian motion.Electron. J. Probab., 1:2 (1996), 1–20 (electronic).

    MathSciNet  Google Scholar 

  21. —, Cut times for simple random walk.Electron. J. Probab., 1:13 (1996), 1–24 (electronic).

    MathSciNet  Google Scholar 

  22. —, The dimension of the frontier of planar Brownian motion.Electron. Comm. Probab., 1:5 (1996), 29–47 (electronic).

    MATH  MathSciNet  Google Scholar 

  23. Lawler, G. F., The frontier of a Brownian path is multifractal. Preprint, 1997.

  24. —, Strict concavity of the intersection exponent for Brownian motion in two and three dimensions.Math. Phys. Electron. J., 4:5 (1998), 1–67 (electronic).

    MathSciNet  Google Scholar 

  25. —, Geometric and fractal properties of Brownian motion and random walk paths in two and three dimensions, inRandom Walks (Budapest, 1998), pp. 219–258. Bolyai Soc. Math. Stud., 9. Janos Bolyai Math. Soc., Budapest, 1999.

    Google Scholar 

  26. Lawler, G. F. &Puckette, E. E., The intersection exponent for simple random walk.Combin. Probab. Comput., 9 (2000), 441–464.

    Article  MathSciNet  Google Scholar 

  27. Lawler, G. F., Schramm, O. &Werner, W., Values of Brownian intersection exponents, I: Half-plane exponents.Acta Math., 187 (2001), 237–273.

    MathSciNet  Google Scholar 

  28. Lawler, G. F., Schramm, O. & Werner, W., Values of Brownian intersection exponents, III: Two-sided exponents. To appear inAnn. Inst. H. Poincaré Probab. Statist. http://arxiv.org/abs/math.PR/0005294.

  29. Lawler, G. F., Schramm, O. & Werner, W., Analyticity of intersection exponents for planar Brownian motion. To appear inActa Math., 188 (2002). http://arxiv.org/abs/math.PR/0005295.

  30. Lawler, G. F., Schramm, O. & Werner, W., Sharp estimates for Brownian non-intersection probabilities. To appear inIn and Out of Equilibrium. Probability with a Physics Flavor. Progr. Probab. Birkhäuser Boston, Boston, MA.

  31. Lawler, G. F. &Werner, W., Intersection exponents for planar Brownian motion.Ann. Probab., 27 (1999), 1601–1642.

    Article  MathSciNet  Google Scholar 

  32. —, Universality for conformally invariant intersection exponents.J. Eur. Math. Soc. (JEMS), 2 (2000), 291–328.

    Article  MathSciNet  Google Scholar 

  33. Löwner, K., Untersuchungen über schlichte konforme Abbildungen des Einheitskreises, I.Math. Ann., 89 (1923), 103–121.

    Article  MATH  MathSciNet  Google Scholar 

  34. Madras, N. &Slade, G.,The Self-Avoiding Walk. Birkhäuser Boston, Boston, MA, 1993.

    Google Scholar 

  35. Mandelbrot, B. B.,The Fractal Geometry of Nature. Freeman, San Francisco, CA, 1982.

    Google Scholar 

  36. Nienhuis, B., Critical behavior of two-dimensional spin models and charge asymmetry in the Coulomb gas.J. Statist. Phys., 34 (1984), 731–761.

    Article  MATH  MathSciNet  Google Scholar 

  37. Pommerenke, Ch., On the Loewner differential equation.Michigan Math. J., 13 (1966), 435–443.

    Article  MATH  MathSciNet  Google Scholar 

  38. Revuz, D. &Yor, M.,Continuous Martingales and Brownian Motion. Grundlehren Math. Wiss., 293. Springer-Verlag, Berlin, 1991.

    Google Scholar 

  39. Schramm, O., Scaling limits of loop-erased random walks and uniform spanning trees.Israel J. Math., 118 (2000), 221–288.

    MATH  MathSciNet  Google Scholar 

  40. Werner, W., Bounds for disconnection exponents.Electron. Comm. Probab., 1:4 (1996), 19–28 (electronic).

    MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Duke University, Box 90320, 27708-0320, Durham, NC, USA

    Gregory F. Lawler

  2. Microsoft Research, 1, Microsoft Way, 98052, Redmond, WA, USA

    Oded Schramm

  3. The Weizmann Institute of Science, Rehovot, Israel

    Oded Schramm

  4. Départment de Mathématiques, Université Paris-Sud, Bât. 425, FR-91405, Orsay Cedex, France

    Wendelin Werner

Authors
  1. Gregory F. Lawler
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  2. Oded Schramm
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  3. Wendelin Werner
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Additional information

The first author was supported by the National Science Foundation.

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Lawler, G.F., Schramm, O. & Werner, W. Values of Brownian intersection exponents, II: Plane exponents. Acta Math. 187, 275–308 (2001). https://doi.org/10.1007/BF02392619

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