the same surface point, so that the number of routes instead exceeds the number of connected surface points. This effects becomes noticeable for $p>0.4$. Figure \[fig:comp\](c) demonstrates the strong connection between the tagged routes and the connections between the midpoint and the surface. This correspondence is further enforced by comparing the size dependence of $r^{\rm tag}$ and $b/B$. For $r^{\rm tag}$ the size dependence follows from Eq. (\[size\]) from which one obtains $r^{\rm tag}\sim \Omega
^{\Phi}$, where $\Omega \sim 2^{L}$ is the size of the system and $\Phi =\ln
\left[\frac{p(1+p)}{1-p(1-p)})\right]/\ln 2$. The exponent $\Phi$ is plotted in Fig. \[fig:comp\](b) and is compared to the corresponding exponent for $b/B\sim \Omega^{\Phi ^{\prime}}$ obtained numerically from simulations [@com; @baek]. As seen $\Phi\approx\Phi^{\prime}$ for $p<0.4$, which again shows that in this interval the number of connected surface points is closely the same as the number of tagged routes. However, as noticed above, for larger values of $p$, the number of routes exceeds the number of connected surface points. This has interesting consequences at the second threshold $p_{c2}$: in Fig. \[fig:comp\](e), the number of tagged routes per surface point is plotted and the curves for the various sizes crosses precisely at $p_{c2}=0.5$. This means that, 