Diskin, Easo Team Solves Percolation Phase

A team of five mathematicians has solved a decades-old puzzle about the speed of phase transitions in networks, proving the so-called 'sharpness conjecture' for percolation on infinite transitive graphs. The breakthrough was achieved in December 2025 by researchers Sahar Diskin, Philip Easo, Ritvik Ramanan Radhakrishnan, Benny Sudakov, and Vincent Tassion at ETH Zurich.

Percolation theory studies how fluids or other influences flow through a network. Its original inspiration was coal. In the 1940s, scientist Rosalind Franklin investigated why some types of coal allowed fluids to pass through while others were impermeable. About a decade later, researchers Simon Broadbent and John Hammersley developed a simple mathematical model to understand carbon filters in gas masks.

Their model imagines a grid of points. 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 distance fluid travels depends on the probability of the coin landing heads.

When this probability is low, fluid collects in small, isolated puddles. Once the probability passes a specific threshold, called the critical probability, the system undergoes a phase transition. The lattice opens up, and fluid can travel extensively. Physicists realized that through this simple percolation model, they could learn about more complex real-world phase transitions, like melting, freezing, and magnetization.

For decades, researchers worked to quantify this phase transition. They wanted to know how quickly pools of fluid grow as the connection probability increases. Many expected that the pools grow very fast-that below the critical probability, puddles are tiny, and above it, a single ocean covers almost everything. This is the sharpness conjecture.

When sharpness was proved for standard lattices in the 1980s, it was foundational. The proof allowed mathematicians to deduce that the flooded portion of the network looks very similar to the underlying lattice. This success led researchers to wonder if sharpness would hold for much broader classes of networks.

In 1996, Itai Benjamini and Oded Schramm began studying percolation on a larger class of graphs called transitive graphs. In a transitive graph, every intersection or point looks the same; there are no local landmarks to distinguish one location from another. A square lattice is one example, but there are many others, including simple loops and infinitely expanding 'trees'.

Over the next decade, Benjamini, Schramm, and colleagues proved that percolation exhibits a phase transition on infinite transitive graphs. However, they still did not know how fast that transition happened. The sharpness conjecture for these graphs was broken 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 of fluid are tiny and far apart.

The 'supercritical' half, dealing with probabilities above the critical threshold, seemed much harder. Here, the landscape should be dominated by one or more infinitely large seas. A proof seemed unattainable, as the existing proof for standard lattices was long, complicated, and could not be adapted.

The new proof by Diskin, Easo, Radhakrishnan, Sudakov, and Tassion finally conquered that challenge. Working intensely in a classroom at ETH Zurich in the week before Christmas 2025, the group glimpsed a simple argument that could handle a huge variety of transitive graphs at once.

They worked through the night to confirm the details, finishing their proof by Christmas. The result confirms that above the critical probability on an infinite transitive graph, large pools of fluid that are not connected to the infinite seas become exceedingly rare. The infinite web is not just a meadow crisscrossed with streams; it is more like an ocean, swamping the entire graph. The proof provides a complete picture of the sharpness of the phase transition across this vast mathematical landscape.