Values of Brownian intersection exponents, II: Plane exponents

By Gregory F. Lawler, Oded Schramm, Wendelin Werner. 2000

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This paper is the follow-up of the paper [27], in which we derived the exact value of intersection exponents between Brownian motions in a half-plane. In the present paper, we will derive the value of intersection exponents between planar Brownian motions (or simple random walks) in the whole plane. This problem is very closely related to the more general question of the existence and value of critical exponents for a wide class of two-dimensional systems from statistical physics, including percolation, self-avoiding walks and other random processes. Theoretical physics predicts that these systems behave in a conformally invariant way in the scaling limit, and uses this fact to predict certain critical exponents associated to these systems. We refer to [27] for a more detailed account on this link and for more references on this subject. Let us now briefly describe some of the results that we shall derive in the present paper. Suppose that B 1, ..., B n are n~>2 independent planar Brownian motions star ted from n different points in the plane. Then it is easy to see (using a subaddit ivity argument) that there exists a constant (n such that

Subfields: Condensed Matter Physics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/58b755fc86bcb63c0e5d70c9871ee5596960b192

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Citations in the corpus

The scaling limits of planar LERW in finitely connected domains. (2006, unknown): “The Annals of Probability 2008, Vol. 36, No. 2, 467–529 DOI: 10.1214/07-AOP342 © Institute of Mathematical Statistics, 2008”
Near-critical percolation in two dimensions (2007, methodology): “Thanks to Smirnov’s proof of conformal invariance of the percolation model at p = 1/2 [46], allowing to prove that critical percolation interfaces converge toward SLE6 as the mesh size goes to zero, and to the derivation of the SLE6 critical exponents [34; 35] by Lawler, Schramm and Werner, it is po”
On the scaling limit of planar self-avoiding walk (2002, background): “A “typical” point on the frontier of a planar Brownian motion looks like a pair of Brownian motions (the “past” and the “future”) starting at the point conditioned so that the union of the paths does not disconnect the point from infinity.”
Hausdorff dimensions for SLE6 (2002, methodology): “It is then possible, using the Feynman–Kac formula (following exactly [8]), to prove that if b > 0,”
Evolution families and the Loewner equation I: the unit disc (2008, methodology): “After that, Schramm [25] and Lawler, Schramm and Werner [15], [16] proved the Mandelbroit conjecture using a stochastic version of this chordal Loewner equation.”
Values of Brownian intersection exponents III: Two-sided exponents (2000, unknown): “We first find the two-sided exponents for the stochastic Loewner evolution processes in a half-plane, from which the Brownian intersection exponents are determined via a universality argument.”
The dimension of the SLE curves (2002, background): “For each value > 0, this de nes a random process denoted by SLE (see e.g. [14] for more details on SLE).”