Finite-Size Scaling in Two-dimensional Continuum Percolation Models

TL;DR

This study investigates the finite-size scaling behavior of cluster mass in two-dimensional continuum percolation models, demonstrating universal scaling properties similar to lattice models and mapping their behavior onto a unified curve.

Contribution

It is the first to compare and unify the scaling behavior of lattice and continuum 2D percolation models near the percolation threshold.

Findings

Scaling expression of mass M versus size L matches lattice models but with a positive slope.

Normalized mass plotted against an effective parameter collapses data across models.

Unified scaling behavior is observed for both isotropic and directed percolation models.

Abstract

We test the universal finite-size scaling of the cluster mass order parameter in two-dimensional (2D) isotropic and directed continuum percolation models below the percolation threshold by computer simulations. We found that the simulation data in the 2D continuum models obey the same scaling expression of mass M to sample size L as generally accepted for isotropic lattice problems, but with a positive sign of the slope in the ln-ln plot of M versus L. Another interesting aspect of the finite-size 2D models is also suggested by plotting the normalized mass in 2D continuum and lattice bond percolation models, versus an effective percolation parameter, independently of the system structure (i.e. lattice or continuum) and of the possible directions allowed for percolation (i.e. isotropic or directed) in regions close to the percolation thresholds. Our study is the first attempt to map the scaling behaviour of the mass for both lattice and continuum model systems into one curve.