Higher Conformal Multifractality

Bertrand Duplantier

Published 2002-07-31Version 1

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.