$p$ values. For smaller $p$, the data shows that $n_b/L$ vanishes as a power law $n_b/L\propto L^{-\alpha}$ with increasing $L$. However, for $p\approx
0.5$, it instead goes to a constant. This change of behavior signals a phase transition. (b) The numerically determined power-law index $\alpha$ plotted as a function of $p$. The dotted line presents a second-order polynomial fit for eyes guidance. Extrapolation for the data points suggests that $\alpha$ vanishes at $p=0.50(1)$. []{data-label="fig:frac"}](fig3b.eps "fig:"){width="60.00000%"}

The tagged route thresholds values for $p_{c1}$ and $p_{c2}$ are by construction upper bounds. The question is then if they also are the exact values. A heuristic argument for the exactness of $p_{c2}$ goes as follows: at the real $p_{c2}$ a unique percolating cluster is formed. The formation of such a cluster means that the probability of finding paths, which takes you to the top, suddenly increases. Such a probability increase is, by the tagged-route argument, predicted to happen at $p=0.5$. This suggests that the tagged-route prediction $p_{c2}\leq 0.5$ can be sharpened to $p_{c2}=0.5$.

In order to further probe the exactness, we numerically calculate the minimum number of backsteps, $n_b$, needed for a nonzero-backstep path to go from the bottom to the top. Figure \[fig:frac\](a) 