Monte Carlo study of correlated continuum percolation: Universality and percolation thresholds.

Abstract

Critical exponents of the continuum-percolation system of two-dimensional distributions of disks in the penetrable concentric shell model are determined by Monte Carlo simulations and by real-space Monte Carlo renormalization-group techniques. In this model, each disk of diameter \ensuremath{\sigma} is composed of a mutually impenetrable core of diameter \ensuremath{\lambda}\ensuremath{\sigma} (0\ensuremath{\le}\ensuremath{\lambda}\ensuremath{\le}1) encompassed by a perfectly penetrable shell of thickness (1-\ensuremath{\lambda})\ensuremath{\sigma}/2. Pairs of particles are connected when the interparticle distance is less than \ensuremath{\sigma}. We find that the susceptibility exponent \ensuremath{\gamma} is given by 2.50\ifmmode\pm\else\textpm\fi{}0.03 for an impenetrability parameter \ensuremath{\lambda}=0.8 and the correlation-length exponent \ensuremath{\nu} to be between 1.30 and 1.35 for various values of \ensuremath{\lambda}. Both results consistently suggest that continuum percolation in the penetrable concentric shell model for nonzero hard-core radii belongs to the same universality class as that of ordinary lattice percolation and of randomly centered disks, as far as the geometrical critical exponents are concerned. We also present the critical reduced number densities and critical area fractions for selected values of \ensuremath{\lambda}.