Many transitive graphs signify objects from different mathematical subfields, like algebra or geometry. Benjamini was intrigued by these interdisciplinary prospects — he hoped the percolation course of would reveal insights into the graph itself. “You might have a stage, which is geometry, and a dancer, which is the random course of,” he stated. By watching the dancer, he hoped to be taught extra in regards to the stage.
Over the subsequent decade, Benjamini, Schramm, and their colleagues printed a flurry of outcomes on the percolation of transitive graphs. They proved that, for a category of infinite transitive graphs, percolation reveals a part transition as you open up edges to stream: Small, remoted swimming pools of fluid all of the sudden coalesce into an infinite internet of linked rivers.
However they nonetheless didn’t know the way quick that transition occurred. Under the important level, how large and the way quite a few had been the swimming pools? Above it, was the infinite internet a meadow crisscrossed with streams — or was it extra like an ocean, swamping all the graph?
Benjamini and Schramm suspected {that a} model of the sharpness conjecture was true on all infinite transitive graphs. That conjecture might be damaged down into two separate issues. The “subcritical” half — addressing what occurs beneath the important level — was accomplished in 2007, by Tonći Antunović and Ivan Veselić. Their work confirmed that right here, swimming pools of fluid are tiny and much aside. Even a hair beneath the important level, the system appears to be like extra like Arizona than Minnesota.
You might have a stage, which is geometry, and a dancer, which is the random course of.
Itai Benjamini, Weizmann Institute for Science
The “supercritical” half of the conjecture — which offers with possibilities above the important threshold — appeared tougher. Right here, the panorama ought to be made up of presumably many seas, every infinitely massive. On this situation, massive swimming pools that aren’t linked to the infinite seas develop into exceedingly uncommon. That’s as a result of a big, remoted pool can solely keep separate if there may be lots of dry land — or closed edges — round it.
However a proof of supercritical sharpness appeared unattainable. For one factor, the earlier work was no assist: A proof of supercritical sharpness on lattices was lengthy and sophisticated, and it couldn’t be tailored to the extra basic case. Whereas different foundational outcomes had been simplified within the final decade, “this was the one remaining fortress,” Nachmias stated.
Mathematicians engaged on this drawback “did some very lovely issues, initiated the idea, picked all of the low-hanging fruit,” Benjamini stated. “After which we began hitting the wall.”
In 2008, as progress on non-lattice percolation slowed, Schramm died at age 46 in a fall whereas mountain climbing. “We misplaced a genius, Oded Schramm, to a tragic accident,” Benjamini stated. “After which we wanted to attend for some new geniuses to return.”
A couple of decade in the past, the sphere started to speed up once more. However proving supercritical sharpness remained tough.
Then, the workforce in Zurich produced a easy proof.
A Sharp Flip
Diskin, Easo, Radhakrishnan, Sudakov, and Tassion didn’t intend to show supercritical sharpness. For many of fall 2025, they had been attempting to know how important likelihood scales with the variety of edges in graphs.
However the 5 mathematicians wished outcomes by the tip of the semester. As that deadline neared, they nonetheless had nothing resembling a proof. So Easo recommended pivoting to sharpness. He, Diskin, and Radhakrishnan made some progress and introduced their outcomes to Sudakov and Tassion. As Tassion took of their work, an thought — maybe an outrageous one — shaped in his thoughts.
He thought that, with some tweaks, their technique may be sturdy sufficient to show sharpness for all infinite transitive graphs. “From there, it was in my head day and evening,” Tassion stated.
“Vincent went loopy with it,” Diskin stated. “I believe he didn’t sleep for 2 weeks at the least.”
It wasn’t solely Tassion. Over these weeks, the collaboration grew to become frenzied. The mathematicians traded concepts continuously, usually texting late at evening. “We actually all had this hunch that there may be one thing to it,” Diskin stated. “We had been half joking in the beginning … possibly the identical thought may resolve this big conjecture. We had been all laughing at one another, however what if, what if?”
Radical Simplicity
Brimming with pleasure, and with the vacations looming, they determined it was time to get severe and write their paper.
To show that a big remoted pool of fluid is unlikely above the important likelihood, the mathematicians assumed they’d such a pool and studied the encircling shoreline. Alongside that shoreline, there have been streams emptying into the pool, however there have been additionally streams that linked again to one of many infinite seas. If these streams coincided anyplace, the mathematicians would have a contradiction — their so-called finite pool of fluid would really be a part of an infinite sea.
Possibly the identical thought may resolve this big conjecture. We had been all laughing at one another, however what if, what if?
Sahar Diskin, ETH Zurich
If the pool was large, the shoreline was lengthy — that means a bigger space the place the pool may connect with one of many infinite seas. The fivesome confirmed that this made it practically unattainable to keep away from the contradiction.
As they hammered out the final particulars of their paper, they all of the sudden noticed that with a easy change, their argument might be drastically improved.
That they had been utilizing a typical approach in likelihood concept known as sprinkling: They put aside a couple of of their open edges, akin to a slight decreasing of the important likelihood. They then regarded for a big pool among the many remainder of the sides and analyzed the open paths round it. Because the set-aside edges had nothing to do with the pool, they might be analyzed independently. That made it simpler to show that, as soon as mixed with the remainder of the graph, they virtually all the time created a path to one of many infinite seas.
However as they talked, they stumble on an unorthodox enchancment to this technique. In the event that they analyzed the sprinkles first, the proof obtained rather a lot easier. What’s extra, it strengthened the argument sufficient that it labored for all infinite transitive graphs. “We had this ping-pong of concepts,” Diskin stated. “Each time you throw concepts one at one other, all of the sudden this wall turns into extra blurry, till it vanishes utterly. Then it’s a bit scary, since you may even have it.”
Lastly, they had been positive they’d proved it: If the likelihood is anyplace above the important threshold, even only a smidge, then fluid covers practically all the transitive graph.
Two months later, they posted a paper. Their argument applies to percolation on any infinite transitive graph. “In the event you zoom into each sentence within the proof, it feels very acquainted and easy, however the best way they put all of it collectively is genuinely novel,” Nachmias stated.
There isn’t a scarcity of unstudied percolation methods that their approach may apply to — like graphs the place the nodes don’t all look an identical, or extra sophisticated fashions that describe freezing water or quantum supplies.
A serious query stays, although: On three-dimensional lattices — the graphs that almost all intently mirror bodily methods — what occurs precisely on the important likelihood? Is there an infinite sea?
The progress on the issue is particularly important to Benjamini, who waited a decade for his expedition to start out up once more. “For the neighborhood, for us, it’s a really deep and significant theorem, and it’s part of the puzzle,” he stated.
Of the proof, Benjamini stated, “it’s a gem. It’s a gem.”
