Scaling relation for determining the critical threshold for continuum percolation of overlapping discs of two sizes

TL;DR

This paper introduces a scaling relation to accurately predict the critical percolation threshold for a mixture of overlapping discs of two sizes, validated against recent Monte Carlo simulations.

Contribution

It proposes a phenomenological scaling equation that captures the effect of size ratio and density on the percolation threshold for two-sized discs.

Findings

Scaling relation agrees with Monte Carlo estimates

Critical threshold depends on a single scaling variable

Model accurately describes percolation behavior for various size ratios

Abstract

We study continuum percolation of overlapping circular discs of two sizes. We propose a phenomenological scaling equation for the increase in the effective size of the larger discs due to the presence of the smaller discs. The critical percolation threshold as a function of the ratio of sizes of discs, for different values of the relative areal densities of two discs, can be described in terms of a scaling function of only one variable. The recent accurate Monte Carlo estimates of critical threshold by Quintanilla and Ziff [Phys. Rev. E, 76 051115 (2007)] are in very good agreement with the proposed scaling relation.