On position-space renormalization group approach to percolation

Abstract

In a position-space renormalization group (PSRG) approach to percolation one calculates the probabilityR(p,b) that a finite lattice of linear sizeb percolates, wherep is the occupation probability of a site or bond. A sequence of percolation thresholdsp _c (b) is then estimated fromR(p _c ,b)=p _c (b) and extrapolated to the limitb→∞ to obtainp _c =p _c (∞). Recently, it was shown that for a certain spanning rule and boundary condition,R(p _c ,∞)=R _c is universal, and sincep _c is not universal, the validity of PSRG approaches was questioned. We suggest that the equationR(p _c ,b)=α, where α isany number in (0,1), provides a sequence ofp _c (b)'s thatalways converges top _c asb→∞. Thus, there is anenvelope from any point inside of which one can converge top _c . However, the convergence is optimal if α=R _c . By calculating the fractal dimension of the sample-spanning cluster atp _c , we show that the same is true aboutany critical exponent of percolation that is calculated by a PSRG method. Thus PSRG methods are still a useful tool for investigating percolation properties of disordered systems.