Conformal Multifractality of Random Walks, Polymers, and Percolation in Two Dimensions

Abstract

Our aim is to derive from conformal invariance the multifractal spectrum of the harmonic measure near a random fractal, such as the frontier of a random walk, i.e., a Brownian motion, a self-avoiding walk, or a percolation cluster. First we consider the related problem of L planar random walks (or Brownian motions) of large time t, starting at neighboring points, and the probability $P_L (t) \approx t^{ - \zeta _L }$ that their paths do not intersect. By a 2D quantum gravity method, i.e., a non linear map onto a random Riemann surface, the former conjecture that $\zeta _N = \frac{1}{{24}}(4L^2 - 1)$ is established. This also applies to the half-plane where ${\tilde\zeta} _{N} = \frac{L}{3}(1 + 2L)$. The non-intersection exponents of unions of independent paths are obtained from generalization of the above formulae to non integer or non rational values of L. In particular, Mandelbrot’s conjecture for the Hausdorff dimension D _H = 4/3 of the frontier of a Brownian path follows from $L = \frac{3}{2}$ as $D_H = 2 - 2\zeta _{3/2}$. The same techniques apply to the harmonic measures (or electrostatic potential, or diffusion field) near a RW or a SAW, or near a critical percolation cluster, whose moments exhibit a multifractal spectrum. The generalized dimensions D (n) as well as the multifractal functions f (α) are derived, and are shown to be all identical for a Brownian motion, a polymer, or a percolation cluster. These are examples of exact conformal multifractality. They are generalized to Potts clusters.