the open cluster and the percolation probability
Once the random medium is in place, the single most informative quantity is the percolation probability theta(p): the chance that the origin lies in an infinite open cluster. It plays the role of the order parameter of the phase transition — the number that is exactly zero on one side of p_c and strictly positive on the other, like the magnetization of a magnet across its Curie point. Studying theta(p) is how we make the vague phrase 'fluid can flow to infinity' into a precise function of p.
Write C = C(0) for the open cluster of the origin and |C| for the number of vertices it contains. The percolation probability is theta(p) = P_p(|C| = infinity), the probability that the origin's cluster is infinite. By translation invariance the origin is not special, so theta(p) is also the density of vertices that belong to some infinite cluster: by the ergodic theorem the fraction of the box [-n,n]^d lying in an infinite cluster converges to theta(p) almost surely. The function theta is non-decreasing in p (more open edges can only help connectivity, made rigorous by the FKG inequality and a coupling), it is right-continuous, and theta(0) = 0, theta(1) = 1. The critical probability is then defined exactly as the threshold p_c = sup{ p : theta(p) = 0 }: below p_c the origin's cluster is almost surely finite, above p_c it has positive probability of being infinite.
Knowing theta(p) > 0 is not the same as knowing the geometry. A subtle and famous open problem is continuity at the critical point: is theta(p_c) = 0, i.e. is there NO infinite cluster exactly at criticality? This is known for d = 2 (Harris, Kesten) and for large d (Hara-Slade, via the lace expansion), but for the physically interesting intermediate dimensions like d = 3 it remains unproven as of the early 2020s. The reason it matters: theta(p_c) = 0 is what makes the transition continuous (second order), and most of the scaling and conformal-invariance picture presupposes it.
On the infinite binary tree (Bethe lattice) the model is exactly solvable: each vertex has 2 children, and the cluster is a Galton-Watson tree with Binomial(2,p) offspring. The origin's cluster is infinite with positive probability exactly when the mean offspring 2p exceeds 1, so p_c = 1/2, and one can compute theta(p) explicitly from the extinction-probability equation.
On a tree, percolation reduces to a branching process — the cleanest case where theta(p) is computable.
theta(p) being positive only tells you an infinite cluster exists with positive probability; by a zero-one (ergodicity) argument that probability is in fact 0 or 1 for the EVENT 'an infinite cluster exists somewhere', but theta(p) = P(origin is in one) is a genuine density strictly between 0 and 1 in the supercritical phase.