Island and lake size distributions in Gradient Percolation

TL;DR

This paper investigates the distribution of island sizes in gradient percolation, revealing power-law behaviors with varying exponents under different gradient conditions and introducing a new gradient bond percolation model.

Contribution

It extends percolation theory by analyzing island size distributions in gradient and nonlinear gradient percolation, including a new bond percolation variant.

Findings

Island size distribution follows a power-law with different exponents from ordinary percolation.

The average island size varies with occupation probability along the gradient.

The nonlinear gradient affects the island size distribution exponent.

Abstract

The well-known problem of gradient percolation has been revisited to study the probability distribution of island sizes. It is observed that as the ordinary percolation, this distribution is also described by a power-law decaying function but the associated critical exponents are found to be different. Because of the underlying gradient for the occupation probability, the average value of the island sizes also has a gradient. The variation of the average island size with the probability of occupation along the gradient has been studied together with its scaling analysis. Further, we have introduced and studied the gradient bond percolation and by studying the island size distribution statistics, we have obtained very similar results. We have also studied the characteristics of the diffusion profile of the particle system on a lattice that is initially half filled and half empty. Here also we observe the same value for the island size probability distribution exponent. Finally, the same study has been repeated for the nonlinear gradient percolation and the value of the island size distribution exponent is found to be a function of the strength of the nonlinear parameter.