Invasion Percolation Between two Sites

TL;DR

This study explores invasion percolation between two sites on a 2D lattice, revealing how local pressure influences cluster size distribution and confirming the process's self-organized criticality and fractal nature.

Contribution

It provides new insights into how local pressure at the extraction site affects cluster statistics and confirms the invariance of fractal dimension in invasion percolation.

Findings

Cluster mass distribution follows a power-law with different exponents depending on pressure.

Fractal dimension of invaded clusters remains consistent regardless of local pressure.

Lattice borders influence cluster statistics and the critical behavior of the process.

Abstract

We investigate the process of invasion percolation between two sites (injection and extraction sites) separated by a distance r in two-dimensional lattices of size L. Our results for the non-trapping invasion percolation model indicate that the statistics of the mass of invaded clusters is significantly dependent on the local occupation probability (pressure) Pe at the extraction site. For Pe=0, we show that the mass distribution of invaded clusters P(M) follows a power-law P(M) ~ M^{-\alpha} for intermediate values of the mass M, with an exponent \alpha=1.39. When the local pressure is set to Pe=Pc, where Pc corresponds to the site percolation threshold of the lattice topology, the distribution P(M) still displays a scaling region, but with an exponent \alpha=1.02. This last behavior is consistent with previous results for the cluster statistics in standard percolation. In spite of these discrepancies, the results of our simulations indicate that the fractal dimension of the invaded cluster does not depends significantly on the local pressure Pe and it is consistent with the fractal dimension values reported for standard invasion percolation. Finally, we perform extensive numerical simulations to determine the effect of the lattice borders on the statistics of the invaded clusters and also to characterize the self-organized critical behavior of the invasion percolation process.