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Values of Brownian intersection exponents. III: Two-sided exponents. (English) Zbl 1006.60075
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.

##### MSC:
 60J65 Brownian motion 81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
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