Values of Brownian intersection exponents. III: Two-sided exponents
type: Item
instance of: scholarly article
title: Values of Brownian intersection exponents. III: Two-sided exponents
zbMATH Open document ID: 1006.60075
DOI: 10.1016/S0246-0203(01)01089-5
author: Gregory F. Lawler
author: Oded Schramm
author: Wendelin Werner
published in: Annales de l'Institut Henri Poincaré. Probabilités et Statistiques
publication date: 2003-02-24T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0005294
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/77706
review text: This is the third in a series of three important papers, which completes the task of determining the intersection exponents of planar Brownian motion \(\xi(n_1,\dots,n_k)\) and \(\widetilde{\xi}(n_1,\dots,n_k)\) for \(n_1,\dots,n_k\) positive integers. [For part I see Acta Math. 187, No. 2, 237-273 (2001; Zbl 1005.60097); for part II see ibid. 187, No. 2, 275-308 (2001; Zbl 0993.60083)]. See the review of part II for an informal definition of intersection exponents with integer arguments. After the work in the previous two parts, the missing exponents are the so-called two-sided exponents. To define the Brownian two-sided exponents \(\widetilde{\xi}(w,1,w)\) in the halfplane for non-integer values of \(w\), let \({\mathfrak B}^k\) be the union of a Brownian path of length \(t\) started in \(\exp(ik\pi/4)\) and the line segment connecting the origin with this starting point. Let \({\mathfrak E}(t)\) be the event that for \(k=1,2,3\) these sets are (except for the origin) disjoint subsets of the upper halfplane. Then \({\mathbb E}[{\mathbb P}({\mathfrak E}(t)\mid {\mathfrak B}^k)^w] \approx t^{-\widetilde{\xi}(w,1,w)}\). Loosely speaking, the added line segments make sure that the three motions maintain their cyclic order around zero. The present paper is devoted to the derivation of the two-sided half plane exponents. The method of proof is very similar to that used in the two companion papers. A crucial role is played by the stochastic Loewner evolution SLE(6). First the two-sided exponents associated to \(\text{SLE}(\kappa)\) in a half-plane are computed. A universality argument allows to derive certain values of the Brownain half-plane exponents and the picture is completed using the cascade relations.
Mathematics Subject Classification ID: 60J65
Mathematics Subject Classification ID: 81T40
zbMATH DE Number: 1751674
zbMATH Keywords: Brownian motion
zbMATH Keywords: intersection exponents
zbMATH Keywords: conformal invariance
zbMATH Keywords: SLE
zbMATH Keywords: two-sided exponents
MaRDI profile type: Publication
arXiv ID: math/0005294
OpenAlex ID: W1974270490
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description: scientific article