Values of Brownian intersection exponents. III: Two-sided exponents
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.
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