Universal crossing of the self-avoiding walk critical exponent nu at the Euclidean value 3/4 for several different fractal families

Abstract

We find that the end-to-end distance critical exponent nu is close to the Euclidean value 3/4 for self-avoiding walks (SAWS) on fractals that are composed of homogeneous parts of size bh approximately=26. More precisely, we find that SAWS on fractals with smaller homogeneous parts (b 3/4, while nu 27. We establish this result in the case of three quite different fractal families: the Sierpinski gasket (SG), family, the plane-filling (PF) family, and the checkerboard (CB) family of fractals. In the case of the first two families (SG and PF), the relevant values for nu were found previously by applying the Monte Carte renormalization group (MCRG) method, but the specific value bh was not recognized. On the other hand, for the CB fractals only the exact renormalization group (RG) results were known previously (for 3<or=b<or=9, where b is an odd integer that serves as the fractal enumerator). In this paper we extend the sequence of the known results for the CB family up to b=81 by generalizing the MCRG method. The new results have revealed the occurrence of the aforementioned crossing of the Euclidean value 3/4. We discuss the significance of the crossing within the current knowledge of SAWS on fractals.