Coastlines and percolation in a model for hierarchical random deposition

TL;DR

This paper analyzes a hierarchical random deposition model, examining coastlines and percolation, revealing a threshold where geometry shifts from Euclidean to fractal and linking coastlines to percolation behavior.

Contribution

It provides a detailed analysis of coastlines and percolation thresholds in a hierarchical deposition model, highlighting the fractal nature at critical points and the relation between coastlines and percolation.

Findings

Number of coastlines exhibits non-universal behavior.

Fractal geometry appears at the critical threshold.

Percolation threshold coincides with the onset of fractal coastlines.

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 $\lambda \, > 1$ 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 $P_c$ 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 $\alpha$. 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 $P_c$ 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.