Five mathematicians settle a decades-old percolation question with one argument for many graphs

The proof's scope is worth explaining in detail. The critical probability, the coin-flip odds above which a lattice stops holding fluid in isolated puddles and suddenly opens up to flow ^[12], is not one number. A square lattice has a different critical probability than a triangular one, and a three-dimensional lattice a different one than a two-dimensional lattice ^[13]. Geometry sets the answer. An argument able to deal with a huge variety of graphs at once ^[6] changes the unit of work in this field, well beyond simply shortening a bibliography.

The lineage is unusually concrete for a piece of probability theory. In the 1940s Rosalind Franklin, then at the British Coal Utilization Research Association, was trying to work out why some coals let fluid pass and others did not; by submerging coal in a variety of fluids she measured the typical size of its holes and how much that size varied ^[2][3]. About a decade later Simon Broadbent and John Hammersley, who wanted to understand the carbon filters in gas masks, wrote down the model ^[10]. Take a lattice of evenly spaced points, flip a coin for each neighboring pair, connect on heads, block on tails, and ask how far the fluid goes ^[11]. Count from Franklin's 1940s measurements to a proof finished in December 2025 and the subject has been open for roughly eight decades ^[17].

What the five answered is a decades-old question about how fast a percolation network floods as you open it up to flow ^[1]. Two of them, Diskin and Easo, were postdocs at the time, and Radhakrishnan was a graduate student ^[5]. The timeline of the last mile is short: convinced by the morning of December 17, proof nailed down by Christmas, eight days ^[7][19]. "We almost didn't believe it at first," Radhakrishnan told Quanta Magazine ^[9]. Asaf Nachmias of Tel Aviv University, who works on percolation and probability, called the proof stunning ^[8].

Now the discipline. Percolation is invoked as a model for the spread of a virus through a city or the propagation of a wildfire ^[16], and a theorem about how quickly a graph gets taken over by one large connected sea ^[14] describes the model itself, separate from any city or any fire. Easo puts the value plainly: rigorous phase transitions in physics are hard, percolation is the caricature, and tools trickle down from it ^[15]. That is the honest claim for this result. It is a better instrument, aimed first at the caricature.

What remains unspecified here is the shape of the fence. Which class of graphs the single argument admits, and how the speed of flooding is quantified, are the load-bearing details, and they live in the write-up rather than in the story of the week before Christmas.