Self-avoiding walks on fractals studied by the Monte Carlo renormalization group

Abstract

The authors have applied the Monte Carlo renormalization group (MCRG) method to study the problem of self-avoiding walks on the Sierpinski gasket family of fractals. Each member of the family is labelled by an integer b, 2<or=b<or= infinity , and when b to infinity both the fractal df and spectral ds dimension approach their Euclidean value 2. They have calculated the critical exponent nu , associated with the mean square end-to-end distance, up to b=80. Their MCRG results deviate at most 0.03% from the available exact results (for 2<or=b<or=9). The obtained data show clearly that nu monotonically decreases with b and crosses the Euclidean value nu =3/4 at b approximately=27, that is, before entering the fractal to Euclidean crossover region that occurs in the limit b to infinity .