Winding angles of long lattice walks

We study the winding angles of random and self-avoiding walks (SAWs) on square and cubic lattices with number of steps N ranging up to 10(7). We show that the mean square winding angle 〈θ(2)〉 of random walks converges to the theoretical form when N → ∞. For self-avoiding walks on the square lattice, we show that the ratio 〈θ(4)〉/〈θ(2)〉(2) converges slowly to the Gaussian value 3. For self-avoiding walks on the cubic lattice, we find that the ratio 〈θ(4)〉/〈θ(2)〉(2) exhibits non-monotonic dependence on N and reaches a maximum of 3.73(1) for N ≈ 10(4). We show that to a good approximation, the square winding angle of a self-avoiding walk on the cubic lattice can be obtained from the summation of the square change in the winding angles of lnN independent segments of the walk, where the ith segment contains 2(i) steps. We find that the square winding angle of the ith segment increases approximately as i(0.5), which leads to an increase of the total square winding angle proportional to (lnN)(1.5).