Computer simulation of trails on a square lattice. I. Trails at infinite temperature

A trail is a random walk on a lattice for which two bonds are not allowed to overlap. However, the chain may cross itself and one may associate with each such intersection an attractive energy ɛ. We study trails at infinite temperature T=∞ (i.e., trails without attractions) on a square lattice using the scanning simulation method. Our results for the radius of gyration and the end-to-end distance strongly suggest (as do previous studies) that the shape exponent is ν=0.75, similar to that for self-avoiding walks (SAW’s). We obtain significantly more accurate estimates than have been obtained before for the entropy exponent γ=1.350±0.012 and for the effective growth parameter μ=2.720 58±0.000 20 (95% confidence limit). The persistence length is found to increase with increasing chain length N and the data fit slightly better an exponential function $N^w$ where w=0.047±0.009 than a logarithmic one. Guttmann [J. Phys. A 18, 567 (1985)] has shown exactly that trails and SAW’s on the hexagonal lattice at T=∞ have the same exponents. Our results suggest that this is true also for the square lattice.