Percolation of randomly distributed growing clusters

TL;DR

This study models the percolation process of growing clusters in two and three dimensions, analyzing how initial seed density influences cluster formation, size distribution, and final coverage, with findings on scaling behavior and dimensional effects.

Contribution

It introduces a model of growing clusters with immediate stopping upon contact, analyzing percolation thresholds and scaling exponents in different dimensions.

Findings

Power law behavior in cluster sizes at low seed density

Final coverage depends on system dimensionality

Scaling exponents for cluster properties are reported

Abstract

We investigate the problem of growing clusters, which is modeled by two dimensional disks and three dimensional droplets. In this model we place a number of seeds on random locations on a lattice with an initial occupation probability, $p$. The seeds simultaneously grow with a constant velocity to form clusters. When two or more clusters eventually touch each other they immediately stop their growth. The probability that such a system will result in a percolating cluster depends on the density of the initially distributed seeds and the dimensionality of the system. For very low initial values of $p$ we find a power law behavior for several properties that we investigate, namely for the size of the largest and second largest cluster, for the probability for a site to belong to the finally formed spanning cluster, and for the mean radius of the finally formed droplets. We report the values of the corresponding scaling exponents. Finally, we show that for very low initial concentration of seeds the final coverage takes a constant value which depends on the system dimensionality.