Winding angles of long 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 $⟨{\theta }^2⟩$ of random walks converges to the theoretical form when $N\to \infty$. For self-avoiding walks on the square lattice, we show that the ratio $⟨{\theta }^4⟩/⟨{\theta }^2{⟩}^2$ converges slowly to the Gaussian value 3. For self avoiding walks on the cubic lattice we find that the ratio $⟨{\theta }^4⟩/⟨{\theta }^2{⟩}^2$ exhibits non-monotonic dependence on $N$ and reaches a maximum of 3.73(1) for $N\approx 10^4$. 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 $\ln N$ 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 $(\ln N{)}^{1.5}$.

Published in The Journal of Chemical Physics

References in corpus (1)

• Accurate estimate of the critical exponent $\nu$ for self-avoiding walks via a fast implementation of the pivot algorithm