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

Five mathematicians at ETH Zurich proved a decades-old theorem about percolation theory, demonstrating how networks abruptly shift behavior past a critical point—a breakthrough that applies to a broad class of graphs and has implications for understanding fluid flow, disease spread, and other network phenomena. The team, working through the holidays in December 2025, discovered a simple yet elegant argument that solved a fundamental question about how quickly percolation networks flood as they open to fluid flow. The proof, which experts have called "stunning," represents a major advance in understanding phase transitions in complex systems.