Mathematicians Crack the Sharpness Conjecture for Percolation

'Stunning' percolation proof solves decades-old puzzle about phase transitions

A team of five mathematicians at ETH Zurich has solved a decades-old problem in percolation theory, proving the sharpness conjecture for all infinite transitive graphs. Their simple proof, completed in December 2025, shows that above the critical probability, a single infinite cluster dominates, resolving a question that had stumped researchers since the 1990s. The result, described as 'stunning' by experts, opens new avenues for understanding phase transitions.