Self-avoiding walks on compact fractals: Exact and Monte Carlo renormalization-group results.

Abstract

We present an exact and Monte Carlo renormalization-group (MCRG) study of the self-avoiding walks (SAW's) on an infinite family of the plane-filling (PF) fractals. The fractals are compact, that is, their fractal dimension ${\mathit{d}}_{\mathit{f}}$ is equal to 2 for all values of the fractal enumerator b (an odd integer, 3\ensuremath{\le}b\ensuremath{\infty}). On the other hand, we demonstrate, via precise calculations, that the corresponding spectral dimension ${\mathit{d}}_{\mathit{s}}$ monotonically increases from 1.393 85 to 2 when b varies from 3 to \ensuremath{\infty}. For the PF fractals, we calculate exactly (for 3\ensuremath{\le}b\ensuremath{\le}9) and through the MCRG approach (for b\ensuremath{\le}121) the SAW critical exponents \ensuremath{\nu} (associated with the mean-squared end-to-end distance) and \ensuremath{\gamma} (associated with the total number of distinct SAW's). The MCRG results for 3\ensuremath{\le}b\ensuremath{\le}9 deviate from exact results at most 0.04% in the case of \ensuremath{\nu} and 0.06% in the case of \ensuremath{\gamma}. Our results show clearly that \ensuremath{\nu} monotonically decreases with b and crosses the Euclidean value \ensuremath{\nu}=3/4 between b=27 and b=29. This is in contrast with all available Flory-type theories, as they predict that \ensuremath{\nu} should be, in the case under study, strictly less than 3/4. In addition, our results show that the critical exponent \ensuremath{\gamma}, being always larger than the Euclidean value 43/32, monotonically increases with b. We discuss, in a framework of the finite-size scaling approach, behavior of \ensuremath{\nu} and \ensuremath{\gamma} in the fractal-to-Euclidean crossover region that occurs when b\ensuremath{\rightarrow}\ensuremath{\infty}. Finally, we discuss a possible relevance of our results to the problem of SAW's on the two-dimensional percolation clusters.