Percolation Proof Solves Decades-Old Puzzle

Percolation Proof Finally Cracks the Sharpness Riddle

Five mathematicians walked into a classroom at ETH Zurich in mid-December 2025 expecting nothing. They left with one of the most sought-after proofs in modern probability theory. Sahar Diskin, Philip Easo, Ritvik Ramanan Radhakrishnan, Benny Sudakov, and Vincent Tassion had been chasing a different problem all semester. With the deadline looming and no results in hand, Easo suggested they pivot toward percolation sharpness, a conjecture that had stumped experts for nearly three decades. What happened next stunned even them. “We almost didn’t believe it at first,” Radhakrishnan said. By the morning of December 17, after working through the night, they knew they had it. By Christmas, the proof was complete. They had answered a decades-old question about how fast a network floods when you open it to fluid flow.

A Problem Born From Coal

The story of percolation starts far from abstract mathematics. In the 1940s, Rosalind Franklin, then working at the British Coal Utilization Research Association, was trying to understand why some types of coal let fluids pass through while others stayed impermeable. Coal is studded with tiny holes, but their size and variation remained a mystery until Franklin measured them by submerging coal in various fluids. About a decade later, Simon Broadbent and John Hammersley formalized her insights into a mathematical model while studying carbon filters in gas masks. The setup is deceptively simple. Take a grid of points, flip a coin for each pair of neighbors, and connect them if it lands on heads. Fluid flows through connected edges. The question: how far does it go? When the probability of connection is low, fluid collects in small, isolated puddles. Above a critical threshold, the lattice suddenly opens up and fluid travels extensively. This is a phase transition, like liquid water turning to ice. Physicists quickly saw the potential. Percolation offers a rigorous way to study melting, freezing, and magnetization without the mess of real-world systems. “Phase transitions in physics are very hard to study rigorously,” Easo said. “Percolation is like the caricature.”

Sharpness and Its Two Halves

The sharpness conjecture predicts that pools of fluid grow extremely fast. Below the critical probability, puddles stay tiny. Above it, a single ocean covers almost everything. Proved on lattices in the 1980s by two independent groups, sharpness became foundational. Mathematicians could deduce the structure of flooded networks from it. But when Itai Benjamini and Oded Schramm extended percolation to transitive graphs in 1996, the question resurfaced. Transitive graphs are networks where every intersection looks the same. A square lattice qualifies, but so do simple loops, infinitely expanding trees, and structures that defy visualization. Many represent objects from algebra and geometry, which excited Benjamini. “You have a stage, which is geometry, and a dancer, which is the random process,” he said. Benjamini and Schramm proved that percolation on infinite transitive graphs exhibits phase transitions. But they couldn't determine how fast those transitions happened. The conjecture split into two halves. The subcritical half, dealing with probabilities below the critical point, fell in 2007 to Tonći Antunović and Ivan Veselić. They showed pools stay tiny and far apart. The supercritical half proved far harder. Above the critical threshold, the landscape should feature infinitely large seas with isolated pools becoming exceedingly rare. But proving this seemed unattainable. The lattice proof was too complicated to adapt, and other foundational results had been simplified, yet this one refused to yield. “This was the one remaining fortress,” said Asaf Nachmias of Tel Aviv University.

An Unexpected Turn

Tragedy struck the field in 2008 when Schramm died at age 46 in a hiking fall. Progress slowed. For years, the problem sat unresolved. Diskin, Easo, Radhakrishnan, Sudakov, and Tassion didn't set out to solve it. Through fall 2025, they focused on how critical probability scales with graph edges. But as the semester's end approached with nothing to show, Easo proposed the pivot. They brought partial results to Sudakov and Tassion. As Tassion absorbed their work, an outrageous idea formed. With tweaks, their strategy might prove sharpness for all infinite transitive graphs. “From there, it was in my head day and night,” Tassion said. “Vincent went crazy with it,” Diskin added. “I think he didn't sleep for two weeks at least.”

Radical Simplicity

The proof's elegance lies in its approach. To show large isolated pools are unlikely above the critical probability, the mathematicians assumed such a pool existed and studied its shoreline. Streams emptied into the pool, but others linked back to infinite seas. If those streams coincided anywhere, the finite pool would actually be part of an infinite sea. A bigger pool means a longer shoreline, creating more opportunities for connection. The five showed this made avoiding the contradiction nearly impossible. They used a technique called sprinkling, setting aside open edges to slightly lower the critical probability. Then came the breakthrough. By analyzing the sprinkles first instead of last, the proof simplified dramatically and strengthened enough to work universally. “We had this ping-pong of ideas,” Diskin said. “Every time you throw ideas one at another, suddenly this wall becomes more blurry, until it vanishes completely.” The final argument applies to percolation on any infinite transitive graph. Even a hair above the critical threshold, fluid covers nearly the entire graph. Nachmias called the proof stunning. “If you zoom into every sentence, it feels very familiar and simple, but the way they put it all together is genuinely novel,” he said.

“For the community, for us, it's a very deep and meaningful theorem, and it's a part of the puzzle,” Benjamini said. “It's a gem. It's a gem.”

What Remains Open

The technique could extend to unstudied percolation systems, including graphs where nodes don't all look identical or models describing freezing water and quantum materials. But one major question persists. On three-dimensional lattices, the graphs closest to physical systems, what happens exactly at the critical probability? Is there an infinite sea? That answer, for now, remains out of reach. For Benjamini, who waited a decade for new geniuses to arrive after Schramm's death, the progress carries deep personal weight. The expedition he started in 1996 has finally resumed. The team posted their paper two months after that December night. The girlfriends and families may not have shared their joy, but Diskin doesn't mind. “It's really rare that you're able to hit something that feels so big and so meaningful,” she said.

Frequently Asked Questions

What problem did the five mathematicians solve, and what was its significance?

They solved the percolation sharpness conjecture, a decades-old question about how fast a network floods when opened to fluid flow. The significance is that they proved that even slightly above the critical threshold, fluid covers nearly the entire graph on any infinite transitive graph.

Who were the five mathematicians that developed the proof, and where did they work?

The five mathematicians were Sahar Diskin, Philip Easo, Ritvik Ramanan Radhakrishnan, Benny Sudakov, and Vincent Tassion. They were at ETH Zurich, as they walked into a classroom there in mid-December 2025.

How did the team come to work on the percolation sharpness problem?

They had been working on a different problem all semester but had no results. As the deadline approached, Easo suggested they pivot toward percolation sharpness, and they brought partial results to Sudakov and Tassion.

What was the key technique that led to the breakthrough, and how did it simplify the proof?

The key technique was sprinkling, where open edges are set aside to slightly lower the critical probability. By analyzing the sprinkles first instead of last, the proof simplified dramatically and strengthened enough to work universally.

When did the team realize they had the proof, and how long did it take to complete?

They realized they had the proof by the morning of December 17, after working through the night. By Christmas, the proof was complete, and they posted their paper two months after that December night.