Conformally invariant fractals and potential theory
The multifractal (MF) distribution of the electrostatic potential near any conformally invariant fractal boundary, like a critical O(N) loop or a Q-state Potts cluster, is solved in two dimensions. The dimension &fcirc;(straight theta) of the boundary set with local wedge angle straight theta is &fcirc;(straight theta) = pi / straight theta-25-c / 12 (pi-straight theta)(2) / straight theta(2pi-straight theta), with c the central charge of the model. As a corollary, the dimensions D(EP) of the external perimeter and D(H) of the hull of a Potts cluster obey the duality equation (D(EP)-1) (D(H)-1) = 1 / 4. A related covariant MF spectrum is obtained for self-avoiding walks anchored at cluster boundaries.
Duplantier
B
Service de Physique Theorique de Saclay, F-91191 Gif-sur-Yvette Cedex, France and Institut Henri Poincare, 11 rue Pierre et Marie Curie, 75231 Paris Cedex 05, France and and Isaac Newton Institute for Mathematical Sciences, 20 Clarkson Road, C.