is a finite probability that that one of the surface points is connected to the midpoint. This condition is given by $\frac{p(1+p)}{1-p(1-p)}=1/2$ and has the solution $p_{c1}=\frac{1}{2}\sqrt{13}-\frac{3}{2}\approx 0.30278$. Since the calculation is based on a particular subclass of possible routes, these results are either upper bounds or exact.

Exactness Argument
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![(Color online) The minimum number of backsteps $n_b$ for a nonzero-backstep path which connects from bottom to the top. (a) The data is plotted as $n_b/L$ against $L$ in a log-log plot for a sequence of fixed $p$ 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"}](fig3a.eps "fig:"){width="60.00000%"} ![(Color online) The minimum number of backsteps $n_b$ for a nonzero-backstep path which connects from bottom to the top. (a) The data is plotted as $n_b/L$ against $L$ in a log-log plot for a sequence of fixed 