Winding angle distribution of lattice trails in two dimensions

The winding angle problem of two-dimensional lattice trails on a square lat tice is studied intensively by the scanning Monte Carlo simulation at infin ite, tricritical, and low-temperatures. The winding angle distribution P-N( theta) and the even moments of winding angle (theta(N)(2k)) are calculated for the lengths of trails up to N = 300. At infinite temperature, trails sh are the same universal winding angle distribution with self-avoiding walks (SAWs), which is a stretched exponential function close to a Gaussian funct ion exp[-theta(2)/ In N] and (theta N-2k) proportional to (In N)(k). Howeve r, trails at tricritical and low-temperatures do not share the same winding angle distribution with SAWs. For trails, PN(theta) is described well by a stretched exponential function exp[-\theta\(alpha)/In N] and (theta(N)(2k) ) proportional to (In N)(2k/alpha) with alpha similar to 1.69 which is far from being a Gaussian and also different from those of SAWs at Theta and lo w-temperatures with alpha similar to 1.54. We provide a consistent numerica l evidence that the winding angle distribution for trails at finite tempera tures may not be a Gaussian function, but, a nontrivial distribution, possi bly a stretched exponential function. Our result also demonstrates that the universality argument between trails and SAWs at infinite and tricritical temperatures indeed persists to the distribution function of winding angle and its associated scaling behavior.