Strict Concavity of the Half Plane Intersection Exponent for Planar Brownian Motion

Gregory F. Lawler (Duke University and Cornell University)


The intersection exponents for planar Brownian motion measure the exponential decay of probabilities of nonintersection of paths. We study the intersection exponent $\xi(\lambda_1,\lambda_2)$ for Brownian motion restricted to a half plane which by conformal invariance is the same as Brownian motion restricted to an infinite strip. We show that $\xi$ is a strictly concave function. This result is used in another paper to establish a universality result for conformally invariant intersection exponents.

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 1-33

Publication Date: March 3, 2000

DOI: 10.1214/EJP.v5-64


  1. L. V. Ahlfors (1973). Conformal Invariants, Topics in Geometric Function Theory McGraw-Hill Math. Review 50:10211
  2. R. Bass (1995). Probabilistic Techniques in Analysis Springer-Verlag Math. Review 96e:60001
  3. P. Berg and J. McGregor (1966). Elementary Partial Differential Equations Holden-Day Math. Review 34:1652
  4. X. Bressaud, R. Fernandez, A. Galves (1999). Decay of correlations for non-Holderian dynamics: a coupling approach Electron. J. Probab. 4 , paper no. 3
  5. B. Duplantier (1999). Two-dimensional copolymers and exact conformal multifractality, Phys. Rev. Lett. 82, 880--883.
  6. G. F. Lawler (1995). Hausdorff dimension of cut points for Brownian motion, Electron. J. Probab. 1, paper no.2. Math. Review 97g:60111
  7. G. F. Lawler (1996). The dimension of the frontier of planar Brownian motion, Electron. Comm. Prob. 1, paper no 5. Math. Review 97g:60110
  8. G. F. Lawler (1997). The frontier of a Brownian path is multifractal, preprint.
  9. G. F. Lawler (1998). Strict concavity of the intersection exponent for Brownian motion in two and three dimensions, Math. Phys. Electron. J. 4, paper no. 5 Math. Review 2000e:60134
  10. G. F. Lawler, W. Werner (1999). Intersection exponents for planar Brownian motion, Ann. Probab. 27, 1601--1642.
  11. G. F. Lawler, W. Werner (1999). Universality for conformally invariant intersection exponents, preprint.

Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.