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

In December 2025, a team of five mathematicians at ETH Zurich made a significant breakthrough in percolation theory, a field studying flow in networks. Percolation captures various phenomena, with fluids flowing through a bed of coffee grounds being a prototypical example. The researchers, including Sahar Diskin, Philip Easo, Ritvik Ramanan Radhakrishnan, Benny Sudakov, and Vincent Tassion, aimed to understand how large connected areas could emerge in graphs, akin to fluid pools.

After working tirelessly, they developed a simple argument applicable to various graphs, confirming its correctness. Their proof answered a decades-old question about the speed at which a percolation network floods as fluid flow is introduced. Percolation has applications in understanding virus spread, gas flow through filters, and wildfire propagation, among other things.

Its origins trace back to Rosalind Franklin's studies of coal properties in the 1940s, which led to a mathematical model by Simon Broadbent and John Hammersley. The model involved a grid of evenly spaced points and a coin flip to determine connections, with fluid flow depending on the probability of the coin landing on heads. Crossing the critical probability threshold resulted in a phase transition, akin to liquid water turning to ice.

The sharpness conjecture, which was proved in 1996, posited that fluid pools would grow very fast above the critical probability, forming a single ocean. The groundbreaking proof by the ETH Zurich team fulfilled this conjecture, providing a deeper understanding of the percolation phase transition.