Intersection Exponents for Planar Brownian Motion

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Intersection Exponents for Planar Brownian Motion

Gregory F. Lawler and Wendelin Werner

Source: Ann. Probab. Volume 27, Number 4 (1999), 1601-1642.

Abstract

We derive properties concerning all intersection exponents for planar Brownian motion and we define generalized exponents that, loosely speaking, correspond to noninteger numbers of Brownian paths. Some of these properties lead to general conjectures concerning the exact value of these exponents.

Primary Subjects: 60J65

Secondary Subjects: 81T40

Keywords: Brownian motion; critical exponents; conformal invariance

Full-text: Open access

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Links and Identifiers

Permanent link to this document: http://projecteuclid.org/euclid.aop/1022874810
Mathematical Reviews number (MathSciNet): MR1742883
Digital Object Identifier: doi:10.1214/aop/1022874810
Zentralblatt MATH identifier: 0965.60071

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References

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

Mathematical Reviews (MathSciNet): MR50:10211

2 AIZENMAN, M. and BURCHARD, A. 1999 . Holder regularity and dimension bounds for ¨ random curves. Duke Math. J. 99 419 453.

Mathematical Reviews (MathSciNet): MR2000i:60012

3 AIZENMAN, M., BURCHARD, A., NEWMAN, C. and WILSON, D. 1999 . Scaling limits for minimal and random spanning trees in two dimensions. Random Structures Algorithms. To appear.

Mathematical Reviews (MathSciNet): MR2001c:60151

4 BURDZY, K. and LAWLER, G. F. 1990 . Non-intersection exponents for random walk and Brownian motion I: Existence and an invariance principle. Probab. Theory Related Fields 84 393 410.

5 BURDZY, K. and LAWLER, G. F. 1990 . Non-intersection exponents for random walk and Brownian motion II: Estimates and applications to a random fractal. Ann. Probab. 18 981 1009.

6 BURDZY, K., LAWLER, G. F. and POLASKI, T. 1989 . On the critical exponent for random walk intersections. J. Statist. Phys. 56 1 12.

Mathematical Reviews (MathSciNet): MR91h:60073

7 BURDZY, K. and WERNER, W. 1996 . No triple point of planar Brownian motion is accessible. Ann. Probab. 24 125 147.

Mathematical Reviews (MathSciNet): MR97j:60147

Zentralblatt MATH: 0860.60063

8 CRANSTON, M. and MOUNTFORD, T. 1991 . An extension of a result by Burdzy and Lawler. Probab. Theory Related Fields 89 487 502.

Mathematical Reviews (MathSciNet): MR92k:60155

9 DUPLANTIER, B. 1992 . Loop-erased self-avoiding walks in two dimensions: exact critical exponents and winding numbers. Phys. A 191 516 522.

10 DUPLANTIER, B. and KWON, K.-H. 1988 . Conformal invariance and intersection of random walks. Phys. Rev. Lett. 61 2514 2517.

11 DUPLANTIER, B., LAWLER, G. F., LE GALL, J.-F. and LYONS, T. J. 1993 . The geometry of the Brownian curve. Bull. Sci. Math. 2 117 91 106.

Mathematical Reviews (MathSciNet): MR93k:60205

Zentralblatt MATH: 0778.60058

12 GODDARD, P. and D. OLIVE, eds. 1988 . Kac Moody and Virasoro Algebras. World Scientific, Singapore.

Mathematical Reviews (MathSciNet): MR89f:17022

13 KENYON, R. 1998 . Conformal invariance of domino tiling. Preprint.

14 KENYON, R. 1998 . The asymptotic determinant of the discrete Laplacian. Preprint.

15 LAWLER, G. F. 1991 . Intersections of Random Walks. Birkhauser, Boston. ¨

16 LAWLER, G. F. 1993 . A discrete analogue of a theorem of Makarov. Combin. Probab. Comput. 2 181 200.

Mathematical Reviews (MathSciNet): MR95c:31004

17 LAWLER, G. F. 1995 . Hausdorff dimension of cut points for Brownian motion. Electron. J. Probab. 1 1 20.

18 LAWLER, G. F. 1996 . The dimension of the frontier of planar Brownian motion. Electron. Comm. Probab. 1 29 47.

Mathematical Reviews (MathSciNet): MR97g:60110

19 LAWLER, G. F. 1997 . Cut times for simple random walk. Electron. J. Probab. 1 1 24.

20 LAWLER, G. F. 1997 . The frontier of a Brownian path is multifractal. Preprint.

21 LAWLER, G. F. 1998 . Strict concavity of the intersection exponent for Brownian motion in two and three dimensions. Math. Phys. Electron. J. 4 1 67.

Mathematical Reviews (MathSciNet): MR2000e:60134

Zentralblatt MATH: 0909.60065

22 LAWLER, G. F. 1999 . Loop-erased random walk. In Perplexing Problems in Probability: Volume in honor of Harry Kesten. Birkhauser, Boston. To appear. ¨

Mathematical Reviews (MathSciNet): MR2000k:60092

Zentralblatt MATH: 0926.60041

23 LAWLER, G. F. and PUCKETTE, E. E. 1997 . The disconnection exponent for simple random walk. Israel J. Math. 99 109 122.

Mathematical Reviews (MathSciNet): MR98h:60107

24 LAWLER, G. F. and PUCKETTE, E. E. 1998 . The intersection exponent for simple random walk. Preprint.

25 LI, B. and SOKAL, A. D. 1990 . High-precision Monte Carlo test of the conformal invariance predictions for two-dimensional mutually avoiding walks. J. Statist. Phys. 61 723 748.

26 MAKAROV, N. G. 1985 . Distortion of boundary sets under conformal mapping. Proc. London Math. Soc. 51 369 384.

27 MANDELBROT, B. B. 1982 . The Fractal Geometry of Nature. Freeman, San Francisco.

Mathematical Reviews (MathSciNet): MR84h:00021

28 PORT, S. C. and STONE, C. J. 1978 . Brownian Motion and Classical Potential Theory. Academic Press, New York.

Mathematical Reviews (MathSciNet): MR58:11459

29 PUCKETTE, E. E. and WERNER, W. 1996 . Monte-Carlo tests and conjectures for disconnection exponents. Electron. Comm. Probab. 1 49 64.

30 REVUZ, D. and YOR, M. 1991 . Continuous Martingales and Brownian Motion. Springer, Berlin.

Mathematical Reviews (MathSciNet): MR92d:60053

31 SCHRAMM, O. 1999 . Scaling limits of loop-erased random walks and uniform spanning trees Israel J. Math. To appear.

32 WERNER, W. 1995 . On Brownian disconnection exponents. Bernoulli 1 371 380.

Zentralblatt MATH: 0845.60083

33 WERNER, W. 1996 . Bounds for disconnection exponents. Electron. Comm. Probab. 1 19 28.

Zentralblatt MATH: 0862.60069

34 WERNER, W. 1997 . Asymptotic behaviour of disconnection and nonintersection exponents. Probab. Theory Related Fields 108 131 152.

35 WILSON, D. B. 1996 . Generating random trees more quickly than the cover time. ACM Symp. Theory of Computing 296 303.

DURHAM, NORTH CAROLINA 27708-0320 91405 ORSAY CEDEX E-MAIL: jose@math.duke.edu FRANCE E-MAIL: wendelin.werner@math.u-psud.fr

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