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Sahar Diskin, Philip Easo, Ritvik Ramanan Radhakrishnan, Benny Sudakov and Vincent Tassion had the proof nailed down by Christmas, and its appeal is that a single argument reaches a huge variety of graphs.
The Scientist · Science desk
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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 [13], 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 [14]. Geometry sets the answer. An argument able to deal with a huge variety of graphs at once [3] 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 [9][10]. About a decade later Simon Broadbent and John Hammersley, who wanted to understand the carbon filters in gas masks, wrote down the model [11]. 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 [12]. Count from Franklin's 1940s measurements to a proof finished in December 2025 and the subject has been open for roughly eight decades [18].
What the five answered is a decades-old question about how fast a percolation network floods as you open it up to flow [4]. Two of them, Diskin and Easo, were postdocs at the time, and Radhakrishnan was a graduate student [2]. The timeline of the last mile is short: convinced by the morning of December 17, proof nailed down by Christmas, eight days [5][19]. "We almost didn't believe it at first," Radhakrishnan told Quanta Magazine [7]. Asaf Nachmias of Tel Aviv University, who works on percolation and probability, called the proof stunning [6].
Now the discipline. Percolation is invoked as a model for the spread of a virus through a city or the propagation of a wildfire [17], and a theorem about how quickly a graph gets taken over by one large connected sea [15] 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 [16]. 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.
Ranked by verification strength, evidence, and original report placement.
They answered a decades-old question about how fast a percolation network floods as you open it up to fluid flow.
In the 1940s Rosalind Franklin was employed at the British Coal Utilization Research Association studying coal, charcoals and graphite; scientists knew coal was studded with tiny holes but did not know why some coals allowed fluids through while others were impermeable.
By submerging coal in a variety of fluids, Franklin was able to measure the typical size of its holes and the amount of variation.
The week before Christmas 2025, five mathematicians (then-postdocs Sahar Diskin and Philip Easo, graduate student Ritvik Ramanan Radhakrishnan, Benny Sudakov and Vincent Tassion) were in a classroom at ETH Zurich perfecting a solution to one of the biggest open problems in percolation theory.
Diskin and Easo were postdocs at the time and Radhakrishnan was a graduate student.
The group had glimpsed a simple argument that could deal with a huge variety of graphs at once.
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Evidence-backed comparisons of source perspectives and observed adoption signals. Read the methodology
Which Builder, Operator, and Investor concerns the observed source mix emphasized—not a truth score.
Evidence, demonstrated adoption, hype gap, incentives, and confidence are assessed independently, each on its own current evidence. How these are measured.
One outlet, no paper to read
The verifiable half of this story is the history: Franklin at BCURA, Broadbent and Hammersley's coin-flip lattice, the shape-dependence of the critical probability, the 1980s sharpness proofs. All of that is textbook and checkable. The new theorem is not — Quanta names the five authors, dates the moment of conviction to the morning of December 17 and quotes Tel Aviv University's Asaf Nachmias praising the result, but points to no manuscript, and the text we have breaks off in the section that sets up the transitive-graph statement.
Nothing to count yet
A proof's uptake looks like other mathematicians using the argument, and our coverage stops at first reactions gathered days after the fact. There is no release, no publication event, no citation, not even a preprint identifier in what we have — so any adoption number would be invented rather than observed.
Superlatives arrive before the citation
"Career-defining", "electric", "stunning" all land in Quanta's opening paragraphs, ahead of anything a reader can inspect. The overreach is mild rather than serious: the dates are precise, the doubt is quoted rather than smoothed away — "we almost didn't believe it at first" — and the claim itself is narrow, about the speed at which a network floods, not about percolation being solved.
No money, but a small field vouching for itself
There is nothing commercial in this story to bend it — no vendor, no funding announcement, no product. What is left is proximity: the outside voices are Nachmias and Itai Benjamini, and the person Quanta asks to weigh the 1980s groundwork is Tom Hutchcroft, whom the piece itself identifies as Easo's doctoral adviser. That is disclosed closeness in a field of a few dozen specialists, which is different from a hidden interest.
Confident about the week, not yet about the theorem
Split the story in two and our confidence splits with it. Who was in the room, when they became convinced, what the field's reaction was: solid, specific, and sourced to named people. Whether the mathematics holds and exactly what class of graphs it covers: unresolved here, with one publisher, no linked write-up and an account that stops before the statement.