Journal of Statistical Physics

, Volume 64, Issue 3–4, pp 829–841 | Cite as

Self-affine fractal clusters: Conceptual questions and numerical results for directed percolation

  • B. Hede
  • J. Kertész
  • T. Vicsek
Short Communications


In this paper we address the question of the existence of a well defined, non-trivial fractal dimensionD of self-affine clusters. In spite of the obvious relevance of such clusters to a wide range of phenomena, this problem is still open since thedifferent published predictions forD have not been tested yet. An interesting aspect of the problem is that a nontrivial global dimension for clusters is in contrast with the trivial global dimension of self-affine functions. As a much studied example of self-affine structures, we investigate the infinite directed percolation cluster at the threshold. We measuredD ind=2 dimensions by the box counting method. Using a correction to scaling analysis, we obtainedD=1.765(10). This result does not agree with any of the proposed relations, but it favorsD=1+(1-σν)/σν, whereν andν are the correlation length exponents andσ is a Fisher exponent in the cluster scaling.

Key words

Fractal dimension self-affinity directed percolation 


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Copyright information

© Plenum Publishing Corporation 1991

Authors and Affiliations

  • B. Hede
    • 1
  • J. Kertész
    • 2
  • T. Vicsek
    • 3
  1. 1.HLRZ c/o KFA JülichJülich 1Germany
  2. 2.Institute for Theoretical Physics, UniversityD-Köln 41Germany
  3. 3.Department of Atomic PhysicsEötvös UniversityBudapestHungary

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