Coastlines and percolation in a model for hierarchical random deposition

Jonas Berx* (Corresponding Author), Evi Bervoets, Claudiu V. Giuraniuc, Joseph O. Indekeu

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

We revisit a known model in which (conducting) blocks are hierarchically and randomly deposited on a d-dimensional substrate according to a hyperbolic size law with the block size decreasing by a factor λ>1 in each subsequent generation. In the first part of the paper the number of coastal points (in D=1) or coastlines (in D=2) is calculated, which are points or lines that separate a region at “sea level” and an elevated region. We find that this number possesses a non-universal character, implying a Euclidean geometry below a threshold value Pc of the deposition probability P, and a fractal geometry above this value. Exactly at the threshold, the geometry is logarithmic fractal. The number of coastlines in D=2 turns out to be exactly twice the number of coastal points in D=1. We comment briefly on the surface morphology and derive a roughness exponent α. In the second part, we study the percolation probability for a current in this model and two extensions of it, in which both the scale factor and the deposition probability can take on different values between generations. We find that the percolation threshold Pc is located at exactly the same value for the deposition probability as the threshold probability of the number of coastal points. This coincidence suggests that exactly at the onset of percolation for a conducting path, the number of coastal points exhibits logarithmic fractal behaviour.
Original languageEnglish
Article number125998
Number of pages14
JournalPhysica A: Statistical Mechanics and its Applications
Volume574
Early online date7 Apr 2021
DOIs
Publication statusPublished - 15 Jul 2021

Keywords

  • Hierarchical deposition
  • Logarithmic fractals
  • Coastlines
  • Percolation
  • Non-universality
  • Coastal point

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