Complexity Explorer Santa Few Institute

Percolation: criticality & the giant component

Percolation theory was conceptualized by Paul Flory and then extended by Walter Stockmeyer, both chemists studying polymers. (Flory would later win the Nobel Prize for his work on macromolecules.) Both were interested in the phase transition from a liquid to a gel of polymers in solution. Through chemical cross-linking, polymers become increasingly interconnected until they behave as a network solid. The Flory-Stockmeyer theory predicts the gel point (phase transition) of the system given the concentrations of a branching units and linear linkage units.

When applying percolation theory to graphs, Flory's and Stockmeyer's critical concentration for polymer gelation can be thought of as the critical probability density of occupied sites in a lattice that allows for a continuous path from one side of the lattice to the opposite side. This spanning path is the giant component. Just as at low polymer concentration, no gel will form, likewise, at low probability density, no giant component will form. Try it out!

Exploring the model

In the default mode of the explorable, increasing the value of p adds more occupied sites to the current configuration. Decreasing the value of p removes occupied sites. All components are colored and the size of the largest component is shown.

You can choose to generate a new configuration everytime you change p.

If you choose `color none` you will see how sites are added/removed when varying p.

If you choose `color spanning`, the critical occupation density can be found by slowly increasing the value of p until a spanning component appears.

More About Percolation

On a square lattice of side length L, each of the N = L x L lattice site is occupied with probability p. when two occupied sites touch, they connect to build a component. If a third site touches one of those two, it belongs to the same component and so forth. With increasing occupation probability p, larger clusters appear.