Effects of disorder on Harris-criterion violating percolation

We present the results of computer simulations on a class of percolative systems, called protected percolation, that violates the Harris criterion. The Harris criterion states whether the critical behavior at a phase transition from a disordered state to an ordered state will be altered by impurities. We incorporate impurities into our simulations to test whether the critical exponents for protected percolation are altered by impurities. We find that the critical exponents for three-dimensional protected percolation simulations indeed change with impurities in the form of missing sites and immortal sites. On the other hand, the critical exponents for both standard percolation and protected percolation in two dimensions are stable against impurities.

Figure 1

Simulation results for the percolation thresholds for standard (circles) and protected (squares) percolation with impurities for a body-centered structure. Missing sites and immortal sites are shown as the top and bottom curves, respectively. The linear terms for all data sets are shown as dashed lines; the solid lines are quadratic fits.

Figure 2

Results of simulations of size ${400}^3$ averaged over 1000 iterations for a body-centered structure of a simulation without impurities compared to the addition of 20% immortal sites. (a) and (c) A log-log plot of the strength of the lattice spanning cluster [see Eqs. (3) and (5)] and (b) and (d) a log-log plot of the second moment of the cluster size distribution [see Eqs. (4) and (6)] for (a) and (b) standard percolation and (c) and (d) protected percolation; the value of the critical exponents changes with the addition of impurities for protected percolation (nonparallel lines), but not in standard percolation (parallel lines) (see Table 1). The gray region highlights the critical region where we determine our exponents through a fitting procedure.