Figure 1

(a) Sites connected by arrows form one of the graphs on which directed percolation can form frozen clusters: e.g., the set of occupied sites (big dots) shown. (b) Site X and its NE, NW, SW, SE pairs of neighbors. (c) Portion of an empty octagonal annulus. The interior of the annulus, as any finite region surrounded by a double frame of vacancies, can be eaten away. Whether the vacant region can expand is determined by the three key sites indicated by question marks. If one of these belongs to an occupied DP path in the NE direction, which is anchored on two perpendicular DP paths as shown, it blocks the expansion. Inset: the full octagon. A necessary condition for the octagon not to be expandable of one step in the NW direction is that a DP path spans the dotted rectangle.Reuse & Permissions

Figure 2

(a) Frozen structure made of DP paths of length $s$ (represented by straight continuous lines) intersecting at $T$ junctions. The A path anchors one end of the B and C paths, while its ends are anchored by the D and E paths. This anchoring is retained if path A is displaced until the nearby dotted line, even if the B and E paths are similarly displaced no further than their corresponding dotted lines. Thus the structure is frozen if all the rectangles—shown and unshown—formed by the solid and dashed lines are spanned lengthwise by DP paths. (b) Two sequences of intersecting rectangles, ${\mathcal{R}}_i$, that, when spanned lengthwise by directed paths, make the central site, $O$, frozen. For clarity, the rectangles in one direction are shown with dashed lines.Reuse & Permissions