Tricritical trails on a square lattice with impenetrable linear boundary: Computer simulation and analytic bounds

Using the scanning simulation method we study self-attracting trails (with energy ɛ per intersection) terminally attached to an impenetrable linear boundary on a square lattice, at their tricritical collapse transition. We obtain (with 95% confidence limits) that the partition function exponents are ${\gamma }_{1t}$=0.634±0.025 (for trails with one end attached to the boundary) and ${\gamma }_{11t}$=-0.44±0.02 (both ends are attached). These results differ significantly from the exact values (8/7≃1.143 and (4/7≃0.571, respectively, derived by Duplantier and Saleur (DS) [Phys. Rev. Lett. 59, 539 (1987)] for self-avoiding walks (SAW’s) on a dilute hexagonal lattice at the FTHETA’ point. Our values are within the error bars of the numerical estimates of Seno and Stella for SAW’s on the same lattice at the FTHETA point. The crossover exponent is calculated in several ways and the various results approximately converge to ${\phi }_t$≃0.71 if corrections to scaling are taken into account. This value is significantly larger than the DS exact value for the FTHETA’ point φ=(3/7≃0.43; it is also larger than recent numerical estimates of ${\phi }_t$ for SAW’s at the FTHETA point. These results suggest that tricritical trails are in a different universality class than SAW’s in both the FTHETA and the FTHETA’ point. The results for the shape exponent ${\nu }_t$ are consistent with the DS value (4/7≃0.571. As expected the results for the tricritical temperature -ε/$k_B$ $T_t$=1.086±0.004 and the growth parameter ${\mu }_t$=2.9902±0.003 are equal to those obtained by the same method for trails on a square lattice without a boundary; these values are slightly smaller than their analytical upper bounds ln3≃1.0986 and 3, respectively.