Two-dimensional continuum percolation models with disks under the generalized Achlioptas process

Abstract

Abstract We investigate the disk percolation phase transitions in continuum models in two dimensions under four kinds of generalized product rule and four kinds of generalized sum rule, respectively. We study the critical behaviors of the largest, second largest clusters and their size ratio to characterize the universality class of percolation transition. Using the finite size scaling analysis and the Monte Carlo simulation, we calculate the critical exponents of the percolation transitions under the above rules. We find that the universality classes of continuum percolation under the generalized product rule, under the generalized sum rule, and the lattice percolation are different.