TY - JOUR

T1 - Coastlines and percolation in a model for hierarchical random deposition

AU - Berx, Jonas

AU - Bervoets, Evi

AU - Giuraniuc, Claudiu V.

AU - Indekeu, Joseph O.

N1 - Publisher Copyright: © 2021

PY - 2021/7/15

Y1 - 2021/7/15

N2 - 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 coastline segments (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 coastline segments 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.

AB - 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 coastline segments (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 coastline segments 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.

KW - Coastal point

KW - Coastlines

KW - Hierarchical deposition

KW - Logarithmic fractals

KW - Non-universality

KW - Percolation

UR - http://www.scopus.com/inward/record.url?scp=85104306249&partnerID=8YFLogxK

U2 - 10.1016/j.physa.2021.125998

DO - 10.1016/j.physa.2021.125998

M3 - Journal article

AN - SCOPUS:85104306249

VL - 574

JO - Physica A: Statistical Mechanics and its Applications

JF - Physica A: Statistical Mechanics and its Applications

SN - 0378-4371

M1 - 125998

ER -