Alternative way to locate the transition temperatures of polymeric models with loops.

We investigate a new criterion that can be used to locate the transition temperatures of walk models and apply the criterion to locate the transition temperatures of trails and silhouettes on a square, a triangular, a simple cubic, and a face-centered-cubic lattice. As the temperature is varied, it is observed that the odd moments or the reduced moments of the persistence lengths undergo a drastic change in a narrow temperature range. The criterion exploits this interesting observation and identifies the drastic change with a collapse of the walk configurations from the swollen phase to the compact phase. The transition temperatures obtained from this criterion for trails on the square, triangular, simple cubic, and face-centered-cubic lattice are, respectively, 1.1, 0.4, 0.6, and 0.3; and for the silhouettes they are, respectively, 1.9, 1.2, 1.4, and 0.9. The results for trails on the square and simple cubic lattices are very close to those of existing computer-simulation results of much longer chains (\ensuremath{\sim}250).