Values of Brownian intersection exponents I: Half-plane exponents

Abstract

This paper proves conjectures originating in the physics literature regarding the intersection exponents of Brownian motion in a half-plane. For instance, suppose that B and B' are two independent planar Brownian motions started from distinct points in a half-plane H. Then, as $t \to \infty$, $$ P [ B[0,t] \cap B'[0,t] = \emptyset, B[0,t] \cup B'[0,t] \subset H ] = t^{-5/3 + o(1)}.$$ The proofs use ideas and tools developped by the authors in previous papers. We prove that one of the stochastic Loewner evolution processes (with parameter 6, that we will call SLE_6 and which has been conjectured to correspond to the scaling limit of critical percolation cluster boundaries) satisfies the ``conformal restriction property''. We establish a generalization of Cardy's formula (for crossings of a rectangle by a percolation cluster) for SLE_6, from which the exact values of intersection exponents for SLE_6 followd. Since this process satisfies the conformal restriction property, the Brownian intersection exponents can be determined from the SLE_6 intersection exponents. Results about intersection exponents in the whole plane appear in subsequent papers.