Multifractal properties of the harmonic measure on Koch boundaries in two and three dimensions

The multifractal properties of the harmonic measure on quadratic and cubic Koch boundaries are studied with the help of a new fast random walk algorithm adapted to these fractal geometries. The conjectural logarithmic development of local multifractal exponents is guessed for regular fractals and checked by extensive numerical simulations. This development allows one to compute the multifractal exponents of the harmonic measure with high accuracy, even with the first generations of the fractal. In particular, the information dimension in the case of the concave cubic Koch surface embedded in three dimensions is found to be slightly higher than its value D1 =2 for a smooth boundary.