top will always contain at least one section where the distance between two backsteps along the path is proportional to $L$. This means that a typical path will always contain sections without backsteps which connects levels separated with a distance which goes to infinity in the limit $L\rightarrow\infty$. The tagged-route argument predicts that such a path exists (does not exist) with a finite probability for $p$ larger (smaller) than $p_{c1}=\frac{1}{2}\sqrt{13}-\frac{3}{2}$. The same argument tells you that $p_{c2}$ cannot be lower than the $p$ value where $\alpha$ vanishes, which according to our numerical extrapolation in Fig. \[fig:frac\](b) gives $0.49< p_{c2} \leq 0.5$. Note that the upper limit in this estimate is $0.5$ because the tagged-route argument by itself gives an upper bound.

To sum up: we conclude that $p_{c1}=\frac{1}{2}\sqrt{13}-\frac{3}{2}$ is the exact value and that $0.49< p_{c2}\leq 0.5$. Furthermore, we conjecture that $p_{c2}=0.5$ is likewise exact based on plausibility arguments.

Comparisons
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![(Color online) Comparisons between $r^{\rm tag}$, the average number of tagged routes from an arbitrary surface point to the midpoint, and, $b$, the number of surface points connected to the midpoint. $r^{\rm tag}$ is given by the analytical solution and $b$ is obtained from numerical simulations. (a) The number 