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

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,...,m} with i≤m. The model is analyzed for all values of i=0,...,7 and m=i,...,10, where z[over ¯](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 z[over ¯](i,m)p_{c}(i,m) is found to converge to a constant as m→∞. Such behavior is expected when i=m (single diamond sizes), for which the product z(i)p_{c}(i) tends toward 2^{d}η_{c}, where η_{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,...,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 the deposition of mixtures of diamonds of radius r=0 and r=1, and contrast it with the (i,m)=(1,2) model and also the continuum percolation of disks of two sizes.