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

We present an exact and Monte Carlo renormalization group (MCRG) study of trails on an infinite family of the plane-filling (PF) fractals, which appeal to be compact, that is, their fractal dimension d(f) is equal to 2 fur all members of the fractal family enumerated by the odd integer b (3 less than or equal to b<infinity). For the PF fractals, we calculate exactly (for 3 less than or equal to b less than or equal to 7) the critical exponents nu, (associated with the mean squared end-to-end distances of trails) and gamma (associated with the total number of different trails). In addition, we calculate nu and gamma through the MCRG approach for b less than or equal to 201 and b less than or equal to 151, respectively. The MCRG results for 3 less than or equal to b less than or equal to 7 deviate from the exact results at most 0.04% in the case of nu, and 0.14% in the case of gamma. Our results show clearly that nu first increases for small values of b (up to b = 9) and then starts to decrease, resembling the large b behavior of nu for self-avoiding walks (SAWs) on the PF fractals. Similarly, our results show that the trail critical exponent 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 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 nu and gamma. [S1063-651X(98)07910-0].