goes all the way to the top for a given $p$ increases. As a consequence, the threshold $p_{c2}$ decreases to a value smaller than 1.

We first make a classification of possible routes to the midpoint from the chosen starting point on the surface. A route classification is defined as an ordered sequence of lattice points, route-markers, given by $(l,d_{l},s_{l})$: if we start from a route-marker point at distance $l$ from the midpoint then the next route-marker point is identified by $d_{l}=\pm 1$, which means that the next is left or right along the horizontal level $l$, together with $s_{l}$ which denotes the distance in lattice points to the next route-marker. At this route marker, the route changes level so that the following route-marker is the closest lattice point on level $l-1$, which means that this point is the first point on the route on level $l-1$. From this first point on $l-1$, we then identify the next two route-markers by $(l-1,d_{l-1},s_{l-1})$, which gives the last point on the route on level $l-1$, as well as the first on level $l-2$. The route classification is illustrated in Fig. \[fig:latt\](c). Next we restrict the routes to all the routes which do 