Mathematicians Solve Percolation Phase

A team of five mathematicians has solved a major open problem in percolation theory, proving how fast a network floods once it passes a critical threshold. The proof, completed in December 2025, confirms the supercritical sharpness conjecture for all infinite transitive graphs. You can explore related mathematical structures in our squad of concepts.

Sahar Diskin, Philip Easo, Ritvik Ramanan Radhakrishnan, Benny Sudakov, and Vincent Tassion worked through the night at ETH Zurich to finalize their argument. "We almost didn't believe it at first," Radhakrishnan told Quanta Magazine. The group had glimpsed a simple argument that could handle a huge variety of graphs simultaneously.

The Physics of Flow and Phase Transitions

Percolation theory studies how fluids or other influences spread through a network. Its original inspiration was coal. In the 1940s, scientist Rosalind Franklin, working for the British Coal Utilization Research Association, investigated why some types of coal allowed fluids to pass through while others were impermeable.

Later, researchers Simon Broadbent and John Hammersley developed a mathematical model to understand carbon filters in gas masks. Their model uses a grid of points, or lattice. For each pair of neighboring points, a coin is flipped. If it lands on heads, the points are connected, allowing flow. If it lands on tails, the connection is blocked.

The system's behavior depends entirely on the probability of a 'heads' result. When the probability is low, fluid collects in small, isolated puddles. Once the probability passes a specific threshold called the critical probability, a phase transition occurs. The lattice opens up, and fluid can travel extensively.

Physicists realized percolation could model other phase transitions, like melting, freezing, and magnetization. "Phase transitions in physics are very hard to study rigorously," Philip Easo said. "Percolation is like the caricature. So people often try to study that first, and then tools trickle down."

The Decades-Long Sharpness Conjecture

For decades, researchers worked to quantify this phase transition. They wanted to know exactly how quickly pools of fluid grow as the connection probability increases. The prevailing expectation, called the sharpness conjecture, stated that the transition is extremely fast. Below the critical probability, puddles remain tiny. Above it, a single giant connected component, or ocean, covers almost the entire network.

This conjecture was proven for standard lattices in the 1980s by two independent groups. Tom Hutchcroft of Princeton University and Caltech, who was Easo's doctoral adviser, called that result "foundational." It allowed mathematicians to deduce that the flooded portion of a lattice network looks very similar to the underlying lattice structure.

However, mathematicians wanted to extend percolation theory to a much broader class of networks known as transitive graphs. In a transitive graph, every intersection or point looks the same from a local perspective. A square lattice is one example, but the class includes many other structures, like simple loops and infinitely expanding trees, some of which are nearly impossible to visualize.

Extending the Proof to New Networks

In 1996, Itai Benjamini and Oded Schramm began exploring percolation on these transitive graphs. They proved that a phase transition occurs on infinite transitive graphs, where small pools suddenly coalesce into an infinite connected web. Yet, the speed of this transition-the sharpness-remained unknown. Benjamini and Schramm suspected a version of the sharpness conjecture held true for all infinite transitive graphs.

The conjecture splits into two parts. The 'subcritical' half, dealing with probabilities below the critical point, was solved in 2007 by Tonći Antunović and Ivan Veselić. Their work showed that in this regime, pools are tiny and far apart. The 'supercritical' half, addressing probabilities above the threshold, seemed much harder. Here, the landscape should be dominated by one or more infinitely large seas, making large isolated pools exceedingly rare.

A proof for lattices existed but was long, complicated, and impossible to adapt to the general case. Asaf Nachmias of Tel Aviv University noted this problem was "the one remaining fortress" in the field. The new work by Diskin, Easo, Radhakrishnan, Sudakov, and Tassion has now stormed that fortress. This breakthrough is a key data point in the field's overall standings.

A Stunning and Joyful Breakthrough

On any infinite transitive graph, once the connection probability exceeds the critical point, the network is almost entirely taken over by one or more giant connected components. Large pools that are not part of these infinite seas become vanishingly rare. This confirms the supercritical sharpness conjecture in full generality.

The mathematicians worked with intense focus to lock down their simple, powerful argument. "I'm not sure the girlfriends and the families were as happy as we were. But we were all very happy at that moment," Sahar Diskin said. "It's really rare that you're able to hit something that feels so big and so meaningful."

Reaction from the community has been celebratory. Asaf Nachmias, who studies percolation theory and probability, stated simply, "I find great joy in this proof. It's stunning." The result provides a fundamental tool for understanding phase transitions across a vast new array of mathematical and physical networks.