Winding angles of long lattice walks

We study the winding angles of random and self-avoiding walks on square and cubic lattices with number of steps $N$ ranging up to $10^7$. We show that the mean square winding angle $\langle\theta^2\rangle$ of random walks converges to the theoretical form when $N\rightarrow\infty$. For self-avoiding walks on the square lattice, we show that the ratio $\langle\theta^4\rangle/\langle\theta^2\rangle^2$ converges slowly to the Gaussian value 3. For self avoiding walks on the cubic lattice we find that the ratio $\langle\theta^4\rangle/\langle\theta^2\rangle^2$ exhibits non-monotonic dependence on $N$ and reaches a maximum of 3.73(1) for $N\approx10^4$. 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 $\ln N$ independent segments of the walk, where the $i$-th segment contains $2^i$ steps. We find that the square winding angle of the $i$-th segment increases approximately as $i^{0.5}$, which leads to an increase of the total square winding angle proportional to $(\ln N)^{1.5}$.