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Values of Brownian intersection exponents. I: Half-plane exponents. (English) Zbl 1005.60097
This outstanding paper is the first one in a series, devoted to some problems concerning planar Brownian motion, arising from two-dimensional models in statistical physics. Namely, the values of Brownian intersection exponents are computed: Let \({\mathbf B}_{i}\), \(i = 1, \ldots , p\), be independent planar Brownian motions (or simple random walks in \({\mathbb Z}^{2}\)), starting from distinct points in the upper half plane \({\mathbf H}\). Then, for \(n \rightarrow +\infty \): \[ {\mathbf P} \left [ {\mathbf B}_{i}[0, n] \cap {\mathbf B}_{j}[0, n] = \emptyset \text{ and } {\mathbf B}_{i}[0, n] \subseteq {\mathbf H}, \forall i \not = j \right ] = n^{-\zeta _{p} + o(1)} \] where \(\zeta _{p} = {{1}\over {6}} p (2 p + 1)\) is the Brownian intersection exponent. The existence of the intersection exponents is readily established. K. Burdzy and G. F. Lawler [Probab. Theory Relat. Fields 84, No. 3, 393–410 (1990; Zbl 0665.60078)] showed that these intersection exponents are the same for the planar Brownian motion and for the simple random walk in \({\mathbb Z}^{2}\). At the heart of the proofs is \(\text{SLE}_{6}\), the stochastic Loewner evolution process with parameter \(6\), which is conjectured to be the scaling limit of two-dimensional critical percolation cluster boundaries. There is a profound relation between this process, planar Brownian motion and conformal invariance. Special instances of this result have been conjectured in the physics literature for a long time; see for example [B. Duplantier and K.-H. Kwon, Phys. Rev. Lett. 61, 2514–2517 (1988)].
The proof of the main theorem uses a combination of ideas from the following papers: O. Schramm [Isr. J. Math. 118, 221–288 (2000; Zbl 0968.60093)]; G. F. Lawler and W. Werner [Ann. Probab. 27, No. 4, 1601–1642 (1999; Zbl 0965.60071) and J. Eur. Math. Soc. (JEMS) 2, No. 4, 291–328 (2000; Zbl 1098.60081)]. Intersection exponents in the half plane play an important role in understanding the corresponding exponents in the whole plane [computed in the second part of this paper [Acta Math. 187, No. 2, 275–308 (2001; Zbl 0993.60083)].
[For part III, see Ann. Inst. Henri Poincaré, Probab. Stat. 38, No. 1, 109–123 (2002; Zbl 1006.60075)].

MSC:
60J65 Brownian motion
30C35 General theory of conformal mappings
82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60G35 Signal detection and filtering (aspects of stochastic processes)
60G17 Sample path properties
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
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