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

Introduction 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.” Franklin’s Fluids 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. By submerging coal in a variety of fluids, Franklin was able to measure the typical size of its holes, as well as the amount of variation. About a decade later, the researchers Simon Broadbent and John Hammersley — wanting to understand the carbon filters in gas masks — developed a mathematical model. Their idea was simple. Take a grid of evenly spaced points (also called a lattice) and a coin. For each pair of neighboring points, flip the coin. If it lands on heads, connect the points with an edge. Fluid can flow between these points. If the coin lands on tails, the flow is blocked. Repeat this procedure for every pair of points. How far does the fluid go? The answer depends on the probability that your coin lands on heads, which can range from 0% to 100%. When the probability is low, fluid can flow through only a few channels, and so it collects in small, isolated puddles. But once the probability passes a threshold called the critical probability, the lattice suddenly opens up. Fluid can travel extensively through the system. The exact value of the critical probability changes depending on the shape of the lattice — a square lattice has a different critical probability than a triangular one, and a 3D lattice has a different critical probability than a 2D one. But as you move above that critical value, you’ll see a phase transition, like liquid water turning to ice. On a finite graph, crossing the critical probability means the network will become dominated by one large sea of fluid. On an infinite graph — an abstraction where the graph extends forever in all directions — one or several infinite seas will dominate. Physicists quickly realized that through percolation, they could learn about melting and freezing, as well as other phase transitions like magnetization. “Phase transitions in physics are very hard to study rigorously,” Easo said. “Percolation is like the caricature. So people often try to study that first, and then tools trickle down.” For scientists who had long been stymied by complicated real-world phase transitions, “it was catching the essence in a very simple setup,” said Itai Benjamini of the Weizmann Institute for Science. “A lot of things that could cloud the issue were removed.” Itai Benjamini, along with his collaborator Oded Schramm, made early progress studying the percolation of transitive graphs. Courtesy of Itai Benjamini For decades, researchers worked to quantify the percolation phase transition. They wanted to know exactly how quickly pools of fluid can grow as you increase the probability that your coin lands on heads. Many predicted that the pools grow very fast — that below the critical probability, puddles are tiny, and above it, a single ocean covers almost everything. This prediction is called the sharpness conjecture. When sharpness was proved on lattices in the 1980s — by two independent groups, one in New Jersey and one in Moscow — it was “foundational,” said Tom Hutchcroft of Princeton University and the California Institute of Technology, who was Easo’s doctoral adviser. Knowing sharpness, mathematicians can deduce a lot about the structure of the flooded portion of the network — in particular, that it looks very similar to the underlying lattice. So when Benjamini and his colleague Oded Schramm plotted an expedition to bring percolation to new types of networks, it seemed natural to wonder if sharpness would go with them. Off the Grid In 1996, Benjamini and Schramm wanted to study percolation in a much larger class of graphs, called transitive graphs. To understand what a transitive graph is, imagine the graph as a network of roads on a flat, desolate landscape. If you want to know where you are on these roads, the only landmarks are the intersections. But if the graph is transitive, all the intersections look similar — to figure out where you are, you’ll need GPS or a compass. A square lattice is one example of a transitive graph: Every intersection consists of four edges meeting at right angles. But there are many kinds of transitive graphs — simple loops (below left) and infinitely expanding “trees” (below right), as well as ones that are almost impossible to visualize. Many transitive graphs represent objects from other mathematical subfields, like algebra or geometry. Benjamini was intrigued by these interdisciplinary possibilities — he hoped the percolation process would reveal insights into the graph itself. “You have a stage, which is geometry, and a dancer, which is the random process,” he said. By watching the dancer, he hoped to learn more about the stage. Over the next decade, Benjamini, Schramm, and their colleagues published a flurry of results on the percolation of transitive graphs. They proved that, for a class of infinite transitive graphs, percolation exhibits a phase transition as you open up edges to flow: Small, isolated pools of fluid suddenly coalesce into an infinite web of connected rivers. But they still didn’t know how fast that transition happened. Below the critical point, how big and how numerous were the pools? Above it, was the infinite web a meadow crisscrossed with streams — or was it more like an ocean, swamping the entire graph? Benjamini and Schramm suspected that a version of the sharpness conjecture was true on all infinite transitive graphs. That conjecture could be broken down into two separate problems. The “subcritical” half — addressing what happens below the critical point — was completed in 2007, by Tonći Antunović and Ivan Veselić. Their work showed that here, pools of fluid are tiny and far apart. Even a hair below the critical point, the system looks more like Arizona than Minnesota. The “supercritical” half of the conjecture — which deals with probabilities above the critical threshold — seemed harder. Here, the landscape should be made up of possibly many seas, each infinitely large. In this scenario, large pools that are not connected to the infinite seas become exceedingly rare. That’s because a large, isolated pool can only stay separate if there is a lot of dry land — or closed edges — around it. But a proof of supercritical sharpness seemed unattainable. For one thing, the previous work was no help: A proof of supercritical sharpness on lattices was long and complicated, and it couldn’t be adapted to the more general case. While other foundational results were simplified in the last decade, “this was the one remaining fortress,” Nachmias said. Mathematicians working on this problem “did some very beautiful things, initiated the theory, picked all the low-hanging fruit,” Benjamini said. “And then we started hitting the wall.” In 2008, as progress on non-lattice percolation slowed, Schramm died at age 46 in a fall while hiking. “We lost a genius, Oded Schramm, to a tragic accident,” Benjamini said. “And then we needed to wait for some new geniuses to come.” About a decade ago, the field began to accelerate again. But proving supercritical sharpness remained difficult. Then, the team in Zurich produced a simple proof. A Sharp Turn Diskin, Easo, Radhakrishnan, Sudakov, and Tassion didn’t intend to prove supercritical sharpness. For most of fall 2025, they were trying to understand how critical probability scales with the number of edges in graphs.