Higher Conformal Multifractality

Abstract

We derive, from conformal invariance and quantum gravity, the multifractal spectrum f(alpha,c) of the harmonic measure (or electrostatic potential, or diffusion field) near any conformally invariant fractal in two dimensions, corresponding to a conformal field theory of central charge c. It gives the Hausdorff dimension of the set of boundary points where the potential varies with distance r to the fractal frontier as r^{alpha}. First examples are a Brownian frontier, a self-avoiding walk, or a percolation cluster. Potts, O(N) models, and the so-called SLE process are also considered. Higher multifractal functions are derived, like the universal function f_2(alpha,alpha') which gives the Hausdorff dimension of the points where the potential jointly varies with distance r as r^{alpha} on one side of the random curve, and as r^{alpha'} on the other. We present a duality between external perimeters of Potts clusters and O(N) loops at their critical point, obtained in a former work, as well as the corresponding duality in the SLE_{kappa} process for kappa kappa'=16.