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, independent 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 behavior of the mass for both lattice and continuum model systems into one curve.
Many-body localization in continuum systems: Two-dimensional bosons