// HACKER NEWS — CYBERSECURITY
'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.
“We almost didn’t believe it at first,” Radhakrishnan said.
They raced to confirm each detail, eager to get their idea down before it shimmered away — and heedless of the holiday. “I’m not sure the girlfriends and the families were as happy as we were. But we were all very happy at that moment,” Diskin said. “It’s really rare that you’re able to hit something that feels so big and so meaningful.”
They worked through the night. 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. “I find great joy in this proof,” said Asaf Nachmias of Tel Aviv University, who studies percolation theory and probability. “It’s stunning.”
From left: Sahar Diskin, Ritvik Ramanan Radhakrishnan, Philip Easo, Vincent Tassion, and Benny Sudakov take a group selfie after completing their paper on supercritical sharpness.
Percolation can describe many kinds of flow: the spread of a virus through a city, gas passing through a filter, or the propagation of a wildfire. But its original inspiration was coal.
In the 1940s, the scientist Rosalind Franklin — now famous for her work on the structure of DNA — was employed at the British Coal Utilization Research Association (BCURA), trying to understand the intricate properties of coal, charcoals, and graphite. Scientists knew that coal was studded with tiny holes, but they didn’t know why some types of coal allowed fluids to pass through them, while others were impermeable.