ON POSITION-SPACE RENORMALIZATION-GROUP APPROACH TO PERCOLATION

In a position-space renormalization group (PSRG) approach to percolati on one calculates the probability R(p, b) that a finite lattice of lin ear size b percolates, where p is the occupation probability of a sire or bond. A sequence of percolation thresholds p(c)(b) is then estimat ed from R(p(c), b)= p(c)(b) and extrapolated to the limit b --> infini ty to obtain p(c) = p(c)(infinity). Recently, it was shown that for a certain spanning rule and boundary condition, R(p(c), infinity) = R(c) is universal, and since p(c) is not universal, the validity of PSRG a pproaches was questioned. We suggest that the equation R(p(c), b) = al pha, where alpha is any number in (0, 1), provides a sequence of p(c)( b)'s that always converges to p(c) as b --> infinity. Thus, there is a n envelope from any point inside of which one can converge to p(c). Ho wever, the convergence is optimal if alpha = R(c). By calculating the fractal dimension of the sample-spanning cluster at p(c), we show that the same is true about any critical exponent of percolation that is c alculated by a PSRG method. Thus PSRG methods are still a useful tool for investigating percolation properties of disordered systems.