Coastlines and percolation in a model for hierarchical random deposition
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Coastlines and percolation in a model for hierarchical random deposition. / Berx, Jonas; Bervoets, Evi; Giuraniuc, Claudiu V.; Indekeu, Joseph O.
In: Physica A: Statistical Mechanics and its Applications, Vol. 574, 125998, 15.07.2021.Research output: Contribution to journal › Journal article › Research › peer-review
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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 -
ID: 371847637