Values of Brownian intersection exponents. I: Half-plane exponents
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 \(n \rightarrow +\infty \): NEWLINE\[NEWLINE{\mathbf P} \left [ {\mathbf B}_{i}[0, n] \cap {\mathbf B}_{j}[0, n] = \emptyset \text{ and } {\mathbf B}_{i}[0, n] \subseteq {\mathbf H}, \forall i \not = j \right ] = n^{-\zeta _{p} + o(1)}NEWLINE\]NEWLINE where \(\zeta _{p} = {{1}\over {6}} p (2 p + 1)\) is the Brownian intersection exponent. The existence of the intersection exponents is readily established. \textit{K. Burdzy} and \textit{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 [\textit{B. Duplantier} and \textit{K.-H. Kwon}, Phys. Rev. Lett. 61, 2514--2517 (1988)]. NEWLINENEWLINEThe proof of the main theorem uses a combination of ideas from the following papers: \textit{O. Schramm} [Isr. J. Math. 118, 221--288 (2000; Zbl 0968.60093)]; \textit{G. F. Lawler} and \textit{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)]. NEWLINENEWLINE[For part III, see Ann. Inst. Henri Poincaré, Probab. Stat. 38, No. 1, 109--123 (2002; Zbl 1006.60075)].
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