How does a coherent structure emerge from many small random occurrences? For instance, at what point are there enough openings in a sponge for a liquid to flow through? Such questions are critical to a long-standing puzzle of percolation theory, a subfield of probability theory focused on the permeability of networks.
More specifically, for years, mathematicians have sought to better understand a threshold that applies to these networks: above this threshold, the odds are high that a particular network is comprised of infinite open connections, while below it, those odds are low. Better characterizing of this transition has been considered a holy grail for the field. As mathematician Benedikt Jahnel of the Technical University of Braunschweig in Germany has puts it, “If someone manages to solve this problem, they’ll probably receive a Fields Medal.”
Among the many experts who have attempted to find this threshold is French mathematician Hugo Duminil-Copin, who himself received a Fields Medal in 2022 for his work on phase transitions in statistical physics—a topic connected to probability theory. But when it came to the percolation theory conjecture, he, like all the rest, failed. And on August 30, 2026, Duminil-Copin expressed concern in an essay on the new blog Proofs and Prompts that AI would likely beat humans to the punch, writing that it would be “only a matter of time before the most famous conjecture in our field … also falls to the bulldozers.”
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Just days after he posted those words, exactly that seems to have happened: the artificial intelligence company Anthropic released a proof of the conjecture generated by a large language model. “The result evoked ambivalent feelings,” Jahnel tells me. Alongside joy that the conjecture had finally been proven, there was a certain disillusionment that the final, crucial step came from an AI.
Mathematical Connections
Percolation theory emerged in 1957, when mathematicians Simon Ralph Broadbent and John Michael Hammersley investigated how different liquids flow through a porous medium. To do this, they modeled the medium as a network in which nodes correspond to holes and edges correspond to the cracks through which liquid moves. You can imagine this as a gigantic system of interconnected pipes that can each be opened and closed.
Percolation theory is concerned with questions of which points are connected and which are not. More specifically, mathematicians want to find out whether a point (often assumed to be the origin) is part of an infinitely extended, connected cluster. Think again of a gigantic system of pipes, each of which is open or closed with a certain probability. If the probability, p, that an individual pipe is open is low, the probability that the point at the origin belongs to an infinite cluster of connected pipes, θ(p), is zero. On the other hand, if the probability, p, is high that a particular pipe is open, θ(p) takes on a larger value. Depending on the network’s geometry, there is a threshold value, pc, above which the probability θ(p) is greater than zero. That all means that the opening probability, pc, marks a transition between a network of limited connectivity and one where infinite connection can exist.
Unfortunately, these pc values are extremely difficult to calculate. In 1980 mathematician Harry Kesten succeeded in finding one for a two-dimensional square lattice, showing that a probability of 1⁄2 marks the boundary between an unconnected and a connected system. But to date, only a few such percolation thresholds have been found. Hardly anyone assumes that exact values for pc can be determined.
Arguably the most important question in percolation theory concerns the transition from a system with only finite clusters to one with infinite clusters. In one- and two-dimensional square lattices, mathematicians have long known that this transition is continuous. The probability θ(p) for a point in an infinitely large cluster does not increase abruptly from zero to a finite number but rather rises continuously with p from zero to one. The same can be demonstrated for generalizations of square grids in high dimensions. In these grids the points have many neighbors, allowing for statistical averaging that is not valid in lower dimensions. And when such networks are considered in eleven dimensions or higher, a continuous transition occurs.
The great puzzle of percolation theory has been determining what this transition looks like for square lattices in the other dimensions: How does θ(p) change in dimensions 3, 4, 5,…, 10 as p increases? As Anthropic’s AI ​​model has now shown, the transition in these cases is also continuous, confirming what the mathematical community had suspected for decades.
The Future of Mathematics
Seeing work that could, in his opinion, garner a human a Fields Medal solved by an AI raises many questions, Jahnel says. For instance, will such prizes exist in the future—and, if so, who will receive them? “Fields Medals have often, though not always, been awarded for proving theorems,” he adds. “Whether a human being will ever make it onto such a list again is questionable.”
Perhaps these awards need to be reimagined—along with mathematics itself. “Mathematicians no longer base their self-confidence and their raison d’être on proving conjectures,” Jahnel says, “but mathematics encompasses so much more,” including formulating conjectures, translating proofs and insights into the human world and incorporating them into textbooks.
Duminil-Copin made a similar point in his essay: “A mathematical question is much more than a theorem waiting to be proved … it is a lighthouse in the night: it illuminates and guides mathematicians in their wanderings, both esthetic and scientific.”
Understanding, Duminil-Copin argued, is not gained solely through theorems and proofs but also through setbacks. In his own career, his attempts to find the holy grail of percolation theory led to other insights, for example. The hope is that these discoveries will not be lost with increasing use of AI models in mathematics.
This article originally appeared in Spektrum der Wissenschaft and was reproduced with permission. It was translated from the original German version with the assistance of artificial intelligence and reviewed by our editors.
