Harmonic measure around a linearly self-similar tree

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.

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