Percolation Theory: Phase Transitions in Random Networks

When does a random network become one?

Take a lattice and independently mark each site — or each bond between neighbouring sites — as "open" with probability p. When p is small, open elements are scattered and form only tiny isolated clusters. As p increases, clusters merge and grow. Percolation theory asks the deceptively simple question: at what point do open elements link together into a cluster large enough to span the entire structure? The answer turns out to be one of the cleanest, sharpest phase transitions in all of statistical physics — and it appears wherever connectivity, conduction, or contagion depends on chance.

The critical threshold p_c

For an infinite lattice there exists a sharp critical value, the percolation threshold p_c, at which a system-spanning cluster first appears. Below p_c, the probability that any given site belongs to an infinite cluster is exactly zero; above it, that probability is strictly positive. The value of p_c depends on the lattice's geometry and dimension, not on any universal constant:

Site percolation, 2D square lattice:      p_c ≈ 0.5927
Bond percolation, 2D square lattice:      p_c  = 0.5   (exact)
Site percolation, 2D triangular lattice:  p_c  = 0.5   (exact)

Above the threshold, the strength of the spanning cluster — the probability P(p) that a given open site belongs to it — rises continuously from zero following a power law, P(p) ~ (p − p_c)^β, where β is a critical exponent.

Scale invariance and universality

Approaching p_c from either side, the system develops a characteristic length scale — the correlation length ξ, the typical size of finite clusters — which diverges as ξ ~ |p − p_c|^(−ν). Exactly at the threshold there is no finite scale at all: clusters of every size coexist, and the spanning cluster becomes a fractal with dimension about 1.896 in two dimensions rather than the integer dimension of a solid region. The critical exponents β and ν are universal — they depend only on the spatial dimension, not on whether the lattice is triangular or square, or whether sites or bonds are used — which is why one model describes gelation in polymers and the fragmentation of communication networks alike.

Why the transition is sharp, not gradual

A common misconception is that percolation switches on gradually. For an infinite system the appearance of the spanning cluster is perfectly sharp: there is zero global connectivity below p_c and certain connectivity above it. The smoother-looking transition seen in finite computer simulations — including this one, capped at a few hundred cells per side — is a finite-size effect, not a property of the true, infinite-lattice transition. Site and bond percolation are also often mistakenly assumed to share one threshold; in fact the two generally differ even on the same lattice, as the table above shows.

Real-world percolation

Because percolation captures the universal mathematics of emergent connectivity, the same threshold behaviour governs the flow of oil and water through porous rock, the conduction of electricity through composite materials once a filler fraction is exceeded, the spread of forest fires and epidemics across a contact network, and the robustness of the internet or a power grid against random node failures — where removing nodes is simply percolation run in reverse.

Frequently asked questions

Try it live

Everything above runs in your browser — open Percolation Theory and drag the occupation probability past p_c to watch the spanning cluster snap into existence, on both site and bond lattices. Nothing is installed, nothing is uploaded.