The problem
In bond percolation on ℤ², each edge is open with probability p independently. Theorem: the critical probability is \(p_{c}\)(ℤ²) = 1/2 — percolation occurs iff p > 1/2.
The Kesten–Harris Percolation Threshold
historicReferences
History & significance
Broadbent and Hammersley introduced percolation in 1957 as a model of fluid flow through disordered media. Harris immediately proved non-existence for p ≤ 1/2 and conjectured existence for p > 1/2. Twenty-three years passed before Harry Kesten completed the proof using a duality argument combined with a differential inequality. Smirnov's 2001 proof of conformal invariance for site percolation on the triangular lattice (Fields Medal content) opened the SLE era.
The resolution (human proof)
Proven by Harry Kesten, 1980 ('On the critical probability of bond percolation in two dimensions'). The key insight was a left-right crossing duality: absence of horizontal crossings at p implies presence of vertical crossings at 1−p. Combined with Harris's earlier result, this pins \(p_{c}\) = 1/2. The theorem founded rigorous percolation theory and led directly to Aizenman–Barsky's sharpness theorem, Russo–Seymour–Welsh theory, and eventually SLE. For d ≥ 3, \(p_{c}\) remains unknown even numerically to better than simulation precision.