Winding angles of long lattice walks

TL;DR

This paper investigates the winding angles of random and self-avoiding walks on square and cubic lattices, demonstrating convergence to theoretical forms and revealing complex scaling behaviors with walk length.

Contribution

It provides extensive numerical analysis of winding angles for large lattice walks, showing convergence properties and proposing a novel summation approach for cubic lattice walks.

Findings

Mean square winding angle converges to theoretical form for large N.

Ratio of moments for self-avoiding walks approaches Gaussian value 3.

Square winding angle scales approximately as (ln N)^{1.5} for cubic lattice walks.

Abstract

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 $(\ln N)^{1.5}$.