Broadbent & Hammersley, 1957 — the minimal model of "when does it join up"
In 1957 Simon Broadbent was working at the British Coal Utilisation Research Association on a very concrete problem: the filter in a gas mask is a lump of porous carbon, and gas crossing it must travel along pores that happen to link up. The pores are randomly placed. He took the question to the mathematician John Hammersley — how dense must the pores be before gas gets from one side to the other?
Hammersley abstracted it into something pure, and named it percolation: on a regular lattice, each position (or each link) is independently open with probability p; adjacent open positions count as connected; when does a path appear that crosses the whole lattice?
The division of labour between this model and the others is unusually clean. It does not care what is travelling — water, current, a virus, a rumour, migrating animals — it cares about one thing only: is the path there. So it is purely topological: no time, no dynamics, no energy. That is both its limitation and the reason it can be carried almost anywhere.
There is no time dimension in the model — every cell draws its lot simultaneously, once; there is no "which opened first". This is routinely misread: percolation describes the connectivity of a static snapshot, not a spreading process. Anyone using it as a "model of spread" has quietly bolted on a time axis the model does not have.
The threshold pc is a property of the lattice, not of the percolation mechanism. A few values worth holding onto: on the 2D square lattice, bond percolation is exactly 1/2 (proved rigorously by Kesten in 1980, and traceable to the square lattice's self-duality); site percolation on the same lattice is about 0.5927 (no closed form — it can only be computed numerically); on the 2D triangular lattice, site percolation is exactly 1/2 again; on the 3D simple cubic lattice, site percolation needs only about 0.3116. More dimensions and more neighbours mean less density is required.
Raise p gradually on a single random field and three stages appear. At low p, a scatter of unrelated blotches. Approaching the threshold, the mid-sized blotches become large and nearly touch each other. Past the threshold, they are stitched together wholesale into one piece.
Plot "share of the lattice taken by the largest cluster" as a curve and it hugs the floor below threshold, lifts its head at the threshold, and climbs from there. That is this model's order parameter — the quantity whose job is to say which phase the system is in, connected or not.
Two further things happen exactly at the threshold and are worth remembering separately. First, the distribution of cluster sizes follows a power law: there is no typical cluster size there. Second, the largest cluster is a fractal — in two dimensions its fractal dimension is 91/48 ≈ 1.896, less than a surface (2) and more than a line (1), meaning it spans the whole lattice while being riddled with holes at every scale. This holds only at the threshold; step off it in either direction and clusters have a definite characteristic size again.
Any question of the form "lots of local connections, and I care whether the whole thing joins up" can take percolation as a first approximation: whether oil and gas can flow out of porous rock; at what loading randomly mixed conductive particles make a plastic conduct; how dense a forest must be before fire crosses the whole of it; how much fragmented habitat must remain before animals can traverse a region; how many nodes random failure must remove before a network falls apart.
Carried onto networks, the lattice becomes an arbitrary graph and the occupation probability becomes a degree distribution. The conclusion becomes: a giant component exists when ⟨k²⟩/⟨k⟩ > 2 (the Molloy–Reed criterion). For a random graph this reduces to "one link per node on average"; for a network with a very uneven degree distribution, a handful of hubs drags the threshold towards zero.
The core prediction is a single falsifiable one: the dependence of global connectivity on local density is not a smooth curve but changes abruptly around one particular value. If what you measure is a smooth proportional relation, your system is not a percolation system.