Percolation · Broadbent & Hammersley 1957

THE PATH OPENS ALL AT ONCE

P  ∝  (ppc)β   β = 536   pc = 0.5927

Each cell is a pore, open with probability p. Neighbouring open pores merge into clusters — the lattice is one realisation of that process. Until a cluster touches both the inlet and the outlet, the bed is closed and nothing spans. The dashed line is pc = 0.5927, the infinite-lattice threshold; β = 5/36 is theory. A filter cake, a packed freezer, a drain screen: flow is not gradual. It is a threshold.