Self-Avoiding Trail model in two-dimensional regular lattices: Study of conformational quantities and relation with the Self-Avoiding Walk model

The Self-Avoiding Trail (SAT) is a random walk model represented by an N-step lattice path, with forbidden step overlaps. The SAT is a variant of the Self-Avoiding Walk (SAW) model, where a site is visited only once. There is a consensus that SAT is in the same universality class of SAW, implying that the scaling behavior of conformational quantities, represented by the exponents ν0 and Δ1, is the same in both models. However, in comparison to SAW, SAT results exhibit most pronounced finite-size effects and inconsistent estimates of ν0, depending on the lattice coordination number and conformational quantity. The scaling behavior of the persistence length, λN, is in disagreement with recent estimates for the SAW model. Also, the influence of coordination number on the lattice-dependent constants requires a better understanding. In this study, we revisit the SAT model with data generated by the pivot algorithm up to N=8000 steps, for the hexagonal, square and triangular lattices. Accordingly, we find compelling evidences that ν0 and Δ1 values are the same for the SAT and SAW models, regardless the lattice and conformational quantity used, while the most pronounced finite-size effect comes from the higher amplitude of the correction to scaling terms. Similarly to SAW, we find the λN convergence to lattice dependent constant λ∞(SAW) for the SAT model. Based on the ratio λ∞(SAW)/λ∞(SAT)∝z/2, where z is the lattice coordination number, we find that the factor z/2 quantifies the relation between SAW and SAT lattice dependent constants.