Comparative study of self-avoiding trails and self-avoiding walks on a family of compact fractals

Abstract

We present an exact and Monte Carlo renormalization group (MCRG) study of trails on an infinite family of the plane-filling (PF) fractals, which appear to be compact, that is, their fractal dimension ${d}_{f}$ is equal to 2 for all members of the fractal family enumerated by the odd integer $b (3<~b<\ensuremath{\infty}).$ For the PF fractals, we calculate exactly (for $3<~b<~7)$ the critical exponents $\ensuremath{\nu}$ (associated with the mean squared end-to-end distances of trails) and $\ensuremath{\gamma}$ (associated with the total number of different trails). In addition, we calculate $\ensuremath{\nu}$ and $\ensuremath{\gamma}$ through the MCRG approach for $b<~201$ and $b<~151,$ respectively. The MCRG results for $3<~b<~7$ deviate from the exact results at most 0.04% in the case of $\ensuremath{\nu}$ and 0.14% in the case of $\ensuremath{\gamma}.$ Our results show clearly that $\ensuremath{\nu}$ first increases for small values of $b$ (up to $b=9)$ and then starts to decrease, resembling the large $b$ behavior of $\ensuremath{\nu}$ for self-avoiding walks (SAWs) on the PF fractals. Similarly, our results show that the trail critical exponent $\ensuremath{\gamma},$ being always larger than the SAW Euclidean value 43/32, monotonically increases with $b$ and for large $b$ displays virtually the same behavior as the corresponding critical exponent $\ensuremath{\gamma}$ for SAWs on the PF fractals. We comment on a possible relevance of the comparative study of the criticality of trails and SAWs on the PF family of fractals to the problem of the uniqueness of the universality class for trails and SAWs on the two-dimensional Euclidean lattices, by discussing the fractal-to-Euclidean crossover behavior of $\ensuremath{\nu}$ and $\ensuremath{\gamma}.$