Exact Determination of the Percolation Hull Exponent in Two Dimensions

Abstract

By mapping the two-dimensional percolation problem on a Coulomb gas, we obtain the exact fractal dimension of the external perimeter (or "hull") of the infinite percolation cluster: $D_H=\frac{7}{4}$, in agreement with numerical estimates and a recent conjecture. We also determine an infinite set of exact exponents associated with various topologies of this hull. We argue finally that the different fractal dimensions observed recently by Grossman and Aharony, who modified the definition of the hull, are all equal to $D_e=\frac{4}{3}$.