Asymptotic behaviour of the number of self-avoiding walks on finitely ramified fractals

Abstract

We study self-avoiding walks (SAWs) on the checkerboard (CB) family of fractals, each member of which can be labelled by an odd integer b (3<or=b< infinity ), so that the fractal dimension df tends to the Euclidean value 2 when b to infinity . By applying the exact renormalization-group method (for b=3,5 and 7), and the Monte Carlo renormalization-group method (for a sequence of fractals with 5<or=b<or=81), we have calculated the critical exponent gamma , associated with the total number of distinct SAWs. It turns out that gamma , being always larger than the corresponding Euclidean value 43/32 increases monotonically with b. In order to learn the asymptotic behaviour of gamma for large b, we have applied the finite-size scaling (FSS) method (based on previous exact results for wedges of the two-dimensional Euclidean lattices), and thereby we have shown that Applying the same method (FSS), we have also demonstrated that the critical exponent nu , associated with the mean-squared end-to-end distance of SAWs, tends to the Euclidean value 3/4 from below, when b to infinity . The obtained results extend, and in a way weld together, the previous studies of SAWs various families of finitely ramified fractals.