Percolation of random compact diamond-shaped systems on the square lattice

TL;DR

This paper investigates site percolation on a square lattice with random diamond-shaped neighborhoods, analyzing how the critical threshold behaves as neighborhood sizes vary, and explores connections to continuum percolation and object size distributions.

Contribution

It introduces a model of percolation with random diamond neighborhoods, analyzes the asymptotic behavior of the critical threshold, and compares discrete and continuum percolation for mixed object sizes.

Findings

Product of average neighbors and critical threshold converges to a constant as maximum neighborhood size increases.

For fixed minimum neighborhood size, the product tends toward the continuum percolation threshold for diamonds.

Mixtures of different-sized diamonds relate to continuum percolation of disks, with specific mappings for monodisperse cases.

Abstract

We study site percolation on a square lattice with random compact diamond-shaped neighborhoods. Each site $s$ is connected to others within a neighborhood in the shape of a diamond of radius $r_s$, where $r_s$ is uniformly chosen from the set $\{i, i+1, \ldots, m\}$ with $i \leq m$. The model is analyzed for all values of $i = 0, \ldots, 7$ and $m = i, \ldots, 10$, where $\overline{z}(i,m)$ denotes the average number of neighbors per site and $p_c(i,m)$ is the critical percolation threshold. For each fixed $i$, the product $\overline{z}(i,m)\,p_c(i,m)$ is found to converge to a constant as $m \to \infty$. Such behavior is expected when $i=m$ (single diamond sizes), for which the product $\overline{z}(i)\,p_c(i)$ tends toward $2^d\eta_c$, where $\eta_c$ is the continuum percolation threshold for diamond-shaped regions or aligned squares in two dimensions ($d=2$). This case is further examined for $i = 1, \ldots, 10$, and the expected convergence is confirmed. The particular case $i = m$ was first studied numerically by Gouker and Family in 1983. We also study the relation to systems of deposited diamond-shaped objects on a square lattice. For monodisperse diamonds of radius $r$, there is a direct mapping to percolation with a diamond-shaped neighborhood of radius $2r+1$, but when there is a distribution of object sizes, there is no such mapping. We study the case of mixtures of diamonds of radius $r=0$ and $r=1$, and compare it to the continuum percolation of disks of two sizes.