initialising lattice…
small clusters → largespanningFig. 1 · 600×600 · seed 13527c
At pc = ½ clusters appear at every length scale at once. This is the knife edge: the correlation length diverges and the picture looks the same however far you zoom.
One of the simplest models in probability, and one of the sharpest.
Take a square grid. For every edge between neighbouring sites, flip a biased coin: keep the edge with probability p, delete it otherwise, independently of every other edge. What remains is a random graph, and the question is simply how big its connected pieces are.
The answer is not gradual. Below pc = ½ every cluster is finite with probability one, and the expected size of the cluster containing any given site is bounded. Above it, an infinite cluster appears, and it is unique. There is no intermediate regime where clusters grow slowly — the transition is sharp, and it happens at exactly one half, a value pinned down by Kesten in 1980 using the self-duality of the square lattice.
What the plate above shows is a finite window on that process, so the edges blur: at p just under ½ a cluster may still cross an 800×800 box by luck, and at p just over it may fail to. Drag the slider slowly through the threshold and the giant component does not fade in — it snaps together out of pieces that were already waiting.
In by size mode each cluster is shaded by the logarithm of how many sites it holds, so a lattice of isolated points reads as an almost blank sheet. The one cluster that reaches all the way across is the only saturated thing on the page.
A weighted union–find with path halving over 718,800 bonds. Because the random field is fixed and raising p only ever opens bonds, each step unions just the bonds that opened since the last one — about a millisecond, against fifteen for a full rebuild at this size. The slider stays a continuous control rather than a form you submit.