Accurate Estimate of the Critical Exponent $\nu$ for Self-Avoiding Walks via a Fast Implementation of the Pivot Algorithm

We introduce a fast implementation of the pivot algorithm for self-avoiding walks, which we use to obtain large samples of walks on the cubic lattice of up to $33\times {10}^6$ steps. Consequently the critical exponent $\nu$ for three-dimensional self-avoiding walks is determined to great accuracy; the final estimate is $\nu =0.587597(7)$. The method can be adapted to other models of polymers with short-range interactions, on the lattice or in the continuum.

Figure 1

$T(N)$ for ${\mathbb{Z}}^2$ and ${\mathbb{Z}}^3$. Note that these estimates were obtained in a separate data run on a different computer from the main experiment, with lengths from $N=2^7-1$ to $N=2^{28}-1$.

Figure 2

Estimates for ${\Delta }_1$.

Figure 3

Estimates for $\nu$ from fits with biased ${\Delta }_1$. We show the envelope of maximum and minimum values of end points of error bars for all observables.