Harmonic measure around a linearly self-similar tree

Abstract

The authors use the concept of random multiplicative processes to help describe and understand the distribution of the harmonic measure on growing fractal boundaries. The Laplacian potential around a linearly self-similar square Koch tree is studied in detail. The multiplicative nature of this potential, and the consequent multifractality of the harmonic measure are discussed. On prefractal stages, the density d mu of the harmonic measure and the corresponding Holder alpha =-ln d mu are well defined along the boundary, except in the folds where the tangent is undefined. A regularization scheme is introduced to eliminate these local effects. They then consider the probability distributions P( alpha ) d alpha of successive stages, and discuss their collapse into an f ( alpha ) curve. Both the left- and right-hand sides of this curve show good convergence. Other studies indicate that, for DLA, the right-hand tail does not converge. A brief comparison is made between the multifractality of these two cases.