Bernoulli Percolation Model

A Bernoulli percolation model is a model of -dimensional percolation theory in which the open or closed status of each relevant area or element is random and independent. Examples include Bernoulli bond percolation and Bernoulli site percolation, as well as Bernoulli models in discrete percolation theory and continuum percolation theory.

Terminology varies across the literature on percolation theory. Some authors use -dimensional Bernoulli percolation specifically for the standard bond percolation model on the regular point lattice . In this convention, each edge is independently open with probability or closed with probability , where

This formulation uses graph theory terminology but is probabilistic in emphasis (Cerf 2006).

For nearest-neighbor Bernoulli bond percolation on , Anthropic (2026) released an AI-generated Lean 4 development claiming that the percolation probability vanishes at the percolation threshold in every dimension . No independent human verification of the proof had been reported as of Sep. 9, 2026.