Conformal invariance, universality, and the dimension of the Brownian frontier

Details Conformal invariance, universality, and the dimension of the Brownian frontier Conformal invariance, universality, and the dimension of the Brownian frontier

2003-04-24

arxiv.org/abs/math/0304369v1

Proceedings of the ICM, Beijing 2002, vol. 3, 63--72

Mathematics

Probability

Scientific paper

This paper describes joint work with Oded Schramm and Wendelin Werner establishing the values of the planar Brownian intersection exponents from which one derives the Hausdorff dimension of certain exceptional sets of planar Brownian motion. In particular, we proof a conjecture of Mandelbrot that the dimension of the frontier is 4/3. The proof uses a universality principle for conformally invariant measures and a new process, the stochastic Loewner evolution ($SLE$), introduced by Schramm. These ideas can be used to study other planar lattice models from statistical physics at criticality. I discuss applications to critical percolation on the triangular lattice, loop-erased random walk, and self-avoiding walk.

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