of tagged routes as a function of $p$ for various sizes $L$. The curves for the different sizes cross at the critical value $p_{c1}=\frac{1}{2}\sqrt{13}-\frac{3}{2}\approx0.3028$, whereas Fig. \[fig:comp\](b) shows the numbers of surface points which are connected to the midpoint, $b$. These values are obtained from numerical simulations as described in Refs. [@baek] and [@com]. These curves also cross at $p_{c1}$ and the independent numerical estimate in Ref. [@nogawa], $p_{c1}=0.304(1)$, is also in excellent agreement with this result. Note that $r^{\rm tag}$ and $b$ are two distinct quantities although closely related. They would be identical if the connection between a surface point and the midpoint did always involve just a single tagged route. As seen in Fig. \[fig:comp\](c), this is true to good approximation in the region $0.3<p<0.4$. For small $p$, there is a tiny deviation because a single tagged route has a small but finite chance of connecting more than one surface point to the midpoint. Thus if a surface point and a midpoint is always connected by precisely one route, then the total number of routes is slightly smaller than the number of connected surface points. On the other hand, for larger $p$, many routes can be connected to 