Site Percolation

Site percolation is a percolation model in discrete percolation theory on a regular point lattice in -dimensional Euclidean space which considers the point lattice vertices as the relevant entities (left figure). The precise mathematical construction for site percolation as a Bernoulli percolation model is as follows.

First, designate each graph vertex of to be independently "open" with probability and closed otherwise. Next, define an open graph path to be any graph path in all of whose vertices are open, and define at the graph vertex the so-called open cluster to be the set of all vertices which may be attained following only open paths from . Write . The main objects of study in the site percolation model are then the percolation probability

(1)

and the critical probability

(2)

where here, is defined to be the product measure

(3)

is the Bernoulli probability measure which assigns whenever is closed and assigns when is open, and is the percolation threshold. Site models for which have infinite connected components almost surely, whereas those for which have none almost surely.

In general, site percolation is considered more general than bond percolation because every Bernoulli bond percolation model on a graph can be reformulated as a site model on its line graph, but a converse reduction does not hold in general. Mixed percolation is considered to be a bridge between the two. Note, too, the existence of several other variants of site percolation; for example, one could drop the assumption of independence to obtain a non-Bernoulli, dependent site model.

Bou-Rabee (2026) reported that nearest-neighbor Bernoulli site percolation on satisfies for every , so there is no infinite open cluster at the percolation threshold. ChatGPT supplied the mathematical argument, with Claude and GLM supporting formalization. As of Oct. 8, 2026, the Lean development had a successful public build and its statement had been compared with the lattice question, but an independent rebuild and independent mathematical review had not been reported (VibeMathed 2026).