Winding angle distribution of lattice trails in two dimensions

The winding angle problem of two-dimensional lattice trails on a square lattice is studied intensively by the scanning Monte Carlo simulation at infinite, tricritical, and low-temperatures. The winding angle distribution P_N(θ) and the even moments of winding angle 〈θ_N^2k〉 are calculated for the lengths of trails up to N = 300. At infinite temperature, trails share the same universal winding angle distribution with self-avoiding walks (SAWs), which is a stretched exponential function close to a Gaussian function exp[-θ²/ln N] and 〈θ_N^2k〉 ∝ (ln N)^k. However, trails at tricritical and low-temperatures do not share the same winding angle distribution with SAWs. For trails, P_N(θ) is described well by a stretched exponential function exp[-|θ|α/ln N] and 〈_N^2k〉 ∝ (ln N)^2k/α with α ∼ 1.69 which is far from being a Gaussian and also different from those of SAWs at Θ and low-temperatures with α ∼ 1.54. We provide a consistent numerical evidence that the winding angle distribution for trails at finite temperatures may not be a Gaussian function, but, a nontrivial distribution, possibly 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.