HARMONIC MEASURE AROUND A LINEARLY SELF-SIMILAR TREE

We 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. We 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.