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

Percolation theory studies how fluids, gases, or other substances flow through networks, such as coffee grounds, filters, or city streets. The concept originated in the 1940s when scientist Rosalind Franklin studied coal’s tiny holes to understand fluid flow through it. In the 1950s, mathematicians Simon Broadbent and John Hammersley formalized the idea using a simple model: a grid of points connected by edges, where each edge is randomly opened or blocked. If the probability of opening an edge is low, fluid remains in small, isolated pockets. Once the probability crosses a critical threshold, fluid suddenly flows extensively, creating large connected areas—a phenomenon called a phase transition, similar to water turning to ice. This critical threshold varies depending on the network’s structure, such as a square grid or a 3D lattice.

A phase transition occurs when a system abruptly changes state, like liquid water freezing into ice. In percolation, this happens when the probability of opening edges in a network crosses a critical value. Below this threshold, fluid pools are small and disconnected, resembling a desert. Above it, the network becomes dominated by one or more large, connected fluid areas, like an ocean. This transition is sharp, meaning even a small increase in probability above the threshold leads to a dramatic change in connectivity. The sharpness conjecture predicted this behavior for all infinite networks, but proving it for complex graphs remained a major challenge for decades.

A transitive graph is a network where every point or intersection looks identical in terms of connections. For example, a square grid is transitive because each intersection has four edges at right angles, making all points indistinguishable. Other transitive graphs include simple loops or infinitely expanding trees. These graphs are studied because they represent abstract mathematical structures from fields like algebra or geometry. In 1996, mathematicians Itai Benjamini and Oded Schramm began exploring percolation on transitive graphs, aiming to understand how fluid flow reveals properties of the underlying network. They proved that phase transitions occur in these graphs but left the sharpness of the transition unresolved.

Progress on proving the sharpness conjecture for transitive graphs stalled after the 2000s. The subcritical half of the conjecture, addressing fluid behavior below the critical threshold, was solved in 2007 by Tonći Antunović and Ivan Veselić, who showed that pools remain tiny and isolated even near the threshold. However, the supercritical half—describing behavior above the threshold—proved far more difficult. The existing proof for simple grids (lattices) was too complex to adapt to transitive graphs. The field lost a key figure in 2008 when Oded Schramm died at age 46. Progress resumed a decade later, but the supercritical sharpness problem remained unsolved until 2025.

In December 2025, five mathematicians—Sahar Diskin, Philip Easo, Ritvik Ramanan Radhakrishnan, Benny Sudakov, and Vincent Tassion—based at ETH Zurich, made a breakthrough while working on a different problem. They realized their approach could address the supercritical sharpness conjecture for all infinite transitive graphs. Over two weeks, they collaborated intensely, often late into the night, refining their ideas. Their proof relied on analyzing the shoreline of large fluid pools above the critical threshold. They showed that the long shoreline made it nearly impossible for such pools to remain isolated, as they would inevitably connect to the infinite fluid seas. The proof was published in March 2026 and applies to any infinite transitive graph.