Values of Brownian intersection exponents, I: Half-plane exponents

Abstract

Theoretical physics predicts that conformal invariance plays a crucial role in the macroscopic behavior of a wide class of two-dimensional models in statistical physics (see, e.g., [4], [6]). For instance, by making the assumption that critical planar percolation behaves in a conformally invariant way in the scaling limit, and using ideas involving conformal field theory, Cardy [7] produced an exact formula for the limit, as N → ∞, of the probability that, in two-dimensional critical percolation, there exists a cluster crossing the rectangle [0, aN] × [0, bN]. Also, Duplantier and Saleur [13] predicted the “fractal dimension” of the hull of a very large percolation cluster. These are just two examples among many such predictions.