Conformal Fractal Geometry & Boundary Quantum Gravity

This article gives a comprehensive description of the fractal geometry of conformally-invariant (CI) scaling curves, in the plane or half-plane. It focuses on deriving critical exponents associated with interacting random paths, by exploiting an underlying quantum gravity (QG) structure, which uses KPZ maps relating exponents in the plane to those on a random lattice, i.e., in a fluctuating metric. This is accomplished within the framework of conformal field theory (CFT), with applications to well-recognized critical models, like O(N) and Potts models, and to the Stochastic Löwner Evolution (SLE). Two fundamental ingredients of the QG construction are relating bulk and Dirichlet boundary exponents, and establishing additivity rules for QG boundary conformal dimensions associated with mutually-avoiding random sets. From these we derive the non-intersection exponents for random walks (RW’s) or Brownian paths, self-avoiding walks (SAW’s), or arbitrary mixtures thereof. The multifractal (MF) function f(α, c) of the harmonic measure (i.e., electrostatic potential, or diffusion field) near any conformally invariant fractal boundary, is given as a function of the central charge c of the associated CFT. A Brownian path, a SAW in the scaling limit, or a critical percolation cluster have identical spectra corresponding to the same central charge c = 0, with a Hausdorff dimension D = sup α f(α; c = 0) = 4/3, which nicely vindicates Mandelbrot’s conjecture for the Brownian frontier dimension. The Hausdorff dimensions DH of a non-simple scaling curve or cluster hull, and DEP of its external perimeter or frontier, are shown to obey the “superuniversal” duality equation (DH − 1)(DEP − 1) = 1 4 , valid for any value of the central charge c. Higher multifractal functions, like the double spectrum f2(α, α; c) of the double-sided harmonic measure, are also considered. The universal mixed MF spectrum f(α, λ; c) describing the local winding rate λ and singularity exponent α of the harmonic measure near any CI scaling curve is given. The fundamental duality which exists between simple and non-simple random paths is established via an algebraic symmetry of the KPZ quantum gravity map. An extended dual KPZ relation is then introduced for the SLE, which commutes with the κ → κ = 16/κ duality for SLEκ. This allows us to calculate the SLE exponents from simple QG rules. These rules are established from the general structure of correlation functions of arbitrary interacting random sets on a random lattice, as derived from random matrix theory. 2000 Mathematics Subject Classification. Primary 60D05; Secondary 81T40, 05C80, 60J65, 60J45, 30C85, 60K35, 82B20, 82B41, 82B43. c ©0000 (copyright holder)