Percolation
Each cell is independently occupied with probability p. Connected clusters form. At a critical threshold, isolated clusters suddenly merge into a single spanning network that bridges the entire lattice. One parameter. A phase transition. Connectivity from randomness.
pc ≈ 0.5927 (square lattice, site percolation)
What is percolation?
Percolation theory studies how connectivity emerges in random systems. Imagine a grid where each cell is independently “open” with some probability p. At low p, you get isolated clusters — small islands of connected cells. At high p, almost everything is connected. The remarkable thing is the transition between these two regimes: it happens suddenly, at a precise critical probability pc. Below pc, no spanning cluster exists. Above it, one always does.
The critical threshold
For site percolation on the square lattice, the critical threshold is pc ≈ 0.5927. This number is not exact — unlike bond percolation on the triangular lattice (where pc = 1/2 exactly), the square lattice site threshold must be computed numerically. At this precise point, the system is poised at a phase transition. The largest cluster is a fractal — it fills neither zero nor all of the lattice, but something in between, with a fractal dimension of about 91/48.
A genuine phase transition
The percolation transition is a second-order (continuous) phase transition, much like the transition between magnetized and demagnetized states in a ferromagnet. Below pc, the probability that a randomly chosen occupied cell belongs to the spanning cluster is zero. Above pc, it rises as (p − pc)β with critical exponent β = 5/36. These exponents are universal — they depend on the dimension of the lattice but not on its microscopic structure. A honeycomb lattice and a square lattice have different pc values but the same critical exponents.
Real-world applications
Percolation models appear everywhere. In porous media, they describe how fluid flows through rock — oil extraction depends on whether connected pathways exist through the pore network. In epidemiology, they model whether a disease can spread across a population: each “bond” is a social contact, and the occupation probability is the transmission rate. Forest fires spread through percolation on trees. Electrical conductivity in composite materials switches on when conductive particles form a spanning network. Even your morning coffee is a percolation process — hot water percolates through a random packing of ground coffee beans.
Connections
Percolation connects to several other experiments in this lab. The network dynamics experiment explores how information spreads through networks — a process closely related to bond percolation. The sandpile experiment demonstrates self-organized criticality, where a system tunes itself to a critical point. In percolation, we tune p manually; in the sandpile, the system finds its own critical state. Both phenomena share deep mathematical structure through the lens of universality and scaling.