Open each square of a grid with probability p. Low p: little islands. Raise p and… nothing, nothing — then at a sharp threshold near 0.59 a single cluster suddenly spans top to bottom, a bridge across the whole world. Slide p through the transition and watch the connection snap into being.
Each site is open with probability p (seeded, so it is reproducible). A flood-fill finds connected clusters of open sites; the system PERCOLATES if one cluster touches both the top and bottom rows. Near the critical p_c≈0.5927 for the 2D square lattice, the spanning probability jumps from ~0 to ~1 over a vanishing window — a phase transition. A fail-loud self-check throws unless, across many seeds, p=0.45 almost never spans while p=0.75 almost always does.
p_c≈0.5927 is the known numerical value for 2D site percolation (no closed form); the sharp jump is a finite-size version of the true transition. One grid size, seeded — the mechanism is exact, the exact threshold is cited.