Figure 30.

The end-to-end topological distance, $\ell \left(N\right)$ (dashed line), and Euclidean one, $r\left(N\right)$ (red line), for an SAW with $N=9$. Here, $\ell \left(9\right)=6$ and $r\left(9\right)=\sqrt{20}\simeq 4.472$, in lattice units, which are consistent with the bounds, $4.47\lesssim \ell \lesssim 6.32$.