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Self-similarity of harmonic measure on DLA

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  • Evertsz, Carl J.G.
  • Mandelbrot, Benoit B.

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.

Suggested Citation

  • Evertsz, Carl J.G. & Mandelbrot, Benoit B., 1992. "Self-similarity of harmonic measure on DLA," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 185(1), pages 77-86.
    • Handle: RePEc:eee:phsmap:v:185:y:1992:i:1:p:77-86
    DOI: 10.1016/0378-4371(92)90440-2
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    2. Dai, Meifeng & Zhang, Cheng & Zhang, Danping, 2014. "Multifractal and singularity analysis of highway volume data," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 407(C), pages 332-340.
    3. Li, Xuewei & Shang, Pengjian, 2007. "Multifractal classification of road traffic flows," Chaos, Solitons & Fractals, Elsevier, vol. 31(5), pages 1089-1094.

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