bond and site percolation
Percolation is the simplest model of a random porous medium and the cleanest place in probability to watch a phase transition happen. Picture the integer lattice Z^d, with vertices (sites) at the points with integer coordinates and edges (bonds) joining nearest neighbours. We make the medium random by either keeping each bond open independently with probability p (and closed with probability 1-p) — this is bond percolation — or keeping each site open with probability p — site percolation. The fundamental question is whether fluid poured in at the origin can flow arbitrarily far: does there exist, almost surely, an infinite connected region of open elements?
Formally, fix a dimension d >= 2 and a parameter p in [0,1]. In bond percolation the probability space is {0,1}^E where E is the edge set of Z^d, with the product Bernoulli(p) measure P_p; a configuration omega assigns to each edge e the value omega(e) = 1 (open) or 0 (closed), independently. In site percolation we instead use {0,1}^V over the vertices. Two open sites are said to be connected if there is a path between them using only open bonds (in bond percolation) or passing only through open sites (in site percolation). The open cluster C(x) of a vertex x is the set of all vertices connected to x. The whole theory studies the geometry of these random clusters as a function of p, and especially how that geometry changes abruptly at a critical value p_c.
The two versions are close cousins but not identical: any bond model can be recoded as a site model on a related graph (the covering or line graph), so site percolation is in a sense more general, but bond percolation on Z^d is usually the easier object to compute with. Their critical values differ — for example p_c(bond) = 1/2 on Z^2 by Kesten's theorem, while p_c(site) on Z^2 is around 0.5927 and not known in closed form. The deep point is that the qualitative picture — a sharp threshold separating a phase with no infinite cluster from a phase with one — is universal across both, across lattices, and across dimensions, even though the precise number p_c is lattice-dependent and almost always non-explicit.
On Z^2 bond percolation, simulate at p = 0.4: clusters are small islands, the origin's cluster typically has a handful of edges. At p = 0.6 a single open cluster snakes across the whole picture and reaches every boundary. Right at p = 0.5 the clusters are scale-free and fractal-looking, with no characteristic size — the signature of criticality.
The same model at three values of p: subcritical (small clusters), critical (fractal), supercritical (an infinite cluster).
Independence across edges (or sites) is the defining assumption — it is what makes Bernoulli percolation tractable. Models with dependence (like the FK random-cluster model behind the Ising/Potts spins) are genuinely harder and behave differently, even though they share the connectivity language.