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Self-similarity of harmonic measure on DLA

Carl J.G. Evertsz and Benoît Mandelbrot

Physica A: Statistical Mechanics and its Applications, 1992, vol. 185, issue 1, 77-86

Abstract: The right-hand side of the ƒ(α) curve of the harmonic measure on DLA is undefined. This does not necessarily imply that the harmonic measure and the DLA geometry are not self-similar. We show for off-lattice DLA that the right-hand tail satisfies a different rescaling rule. This Cauchy rescaling is compatible with self-similarity. The analysis is done on off-off-lattice DLA in which both the Brownian motion and the Laplace equation are off-lattice. The cluster sizes range between 32 and 50 000 atoms. The square lattice used to numerically estimate the Laplacian potential introduces a lower cutoff on the spatial resolution of this potential. We find a dependence of the right tail of the distribution of Hölders α on this ultraviolet cutoff. Whereas the shape of the tail does depend on this ultraviolet lattice cutoff, the applicability of the collapse rules do not.

Date: 1992
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DOI: 10.1016/0378-4371(92)90440-2

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