‘Stunning’ Percolation Proof Solves Decades-Old Puzzle About Phase Transitions

The week before Christmas 2025, five mathematicians were holed up in a classroom at ETH Zurich. The mood was electric: They were this close to a career-defining breakthrough. The group — consisting of then-postdocs Sahar Diskin and Philip Easo, graduate student Ritvik Ramanan Radhakrishnan, Benny Sudakov, and Vincent Tassion — was perfecting a solution to one of the biggest open problems in percolation theory, the study of flow in a network.

Percolation captures a vast array of phenomena, but the prototypical examples involve fluids, like hot water seeping through a bed of coffee grounds. Diskin, Easo, Radhakrishnan, Sudakov, and Tassion were trying to work out something fundamental about how graphs — networks of points connected by lines, or edges — can be taken over by large connected areas, the equivalent of pools of fluid. The group had glimpsed a simple argument that could deal with a huge variety of graphs at once.

They raced to confirm each detail, eager to get their idea down before it shimmered away — and heedless of the holiday. But we were all very happy at that moment,” Diskin said. By the morning of December 17, exhilarated from the effort, they were convinced their idea was correct.

By Christmas, they’d nailed down a proof. They had answered a decades-old question about how fast a percolation network floods as you open it up to fluid flow.