Percolation

Between p = 0.58 and p = 0.60 almost nothing changes in quantity, yet everything changes in kind. Each cell of this 96×64 grid is opened independently with probability p, adjacent open cells merge into clusters, and right at the threshold p_c ≈ 0.5927 for the two-dimensional square lattice a cluster suddenly appears linking the top edge to the bottom. Exactly at that point the giant cluster is riddled with holes like a fractal and at its most fragile; the larger the grid, the steeper the jump from sealed to leaking.

▸ Building a 96×64 grid — each cell open/closed independently…

▸ Seeding open cells with probability p

▸ Flood-filling to find connected clusters (4-neighbours)

▸ Detecting a spanning cluster linking top ↔ bottom

▸ Locating the critical threshold p_c ≈ 0.5927…

▸ Ready — Online. ✅

Observation panel

💧 Trạng thái

thumbnail of the whole grid

Open probability p—

Open-cell fraction—

Largest cluster—

… of the grid—

Spanning?—

Clusters—

The idea

Gradually raise the open probability p of cells: below threshold only scattered small clusters; past p_c ≈ 0,593 a spanning cluster suddenly emerges linking both edges — the system PERCOLATES. One jump, one phase transition.

—

Purple = open cell · dark = closed cell · orange = largest cluster · green = spanning cluster · drag the p slider past p_c≈0.593 to see the system percolate

Your browser has canvas disabled.

Percolation probability vs p (S-curve) percolationp_c