GoKawiilMany transitive graphs represent objects from other mathematical subfields, like algebra or geometry. Benjamini was intrigued by these interdisciplinary possibilities — he hoped the percolation process would reveal insights into the graph itself. “You have a stage, which is geometry, and a dancer, which is the random process,” he said. By watching the dancer, he hoped to learn more about the stage.
Over the next decade, Benjamini, Schramm, and their colleagues published a flurry of results on the percolation of transitive graphs. They proved that, for a class of infinite transitive graphs, percolation exhibits a phase transition as you open up edges to flow: Small, isolated pools of fluid suddenly coalesce into an infinite web of connected rivers.
But they still didn’t know how fast that transition happened. Below the critical point, how big and how numerous were the pools? Above it, was the infinite web a meadow crisscrossed with streams — or was it more like an ocean, swamping the entire graph?
Benjamini and Schramm suspected that a version of the sharpness conjecture was true on all infinite transitive graphs. That conjecture could be broken down into two separate problems. The “subcritical” half — addressing what happens below the critical point — was completed in 2007, by Tonći Antunović and Ivan Veselić. Their work showed that here, pools of fluid are tiny and far apart. Even a hair below the critical point, the system looks more like Arizona than Minnesota.
You have a stage, which is geometry, and a dancer, which is the random process. Itai Benjamini, Weizmann Institute for Science
The “supercritical” half of the conjecture — which deals with probabilities above the critical threshold — seemed harder. Here, the landscape should be made up of possibly many seas, each infinitely large. In this scenario, large pools that are not connected to the infinite seas become exceedingly rare. That’s because a large, isolated pool can only stay separate if there is a lot of dry land — or closed edges — around it.
But a proof of supercritical sharpness seemed unattainable. For one thing, the previous work was no help: A proof of supercritical sharpness on lattices was long and complicated, and it couldn’t be adapted to the more general case. While other foundational results were simplified in the last decade, “this was the one remaining fortress,” Nachmias said.
Mathematicians working on this problem “did some very beautiful things, initiated the theory, picked all the low-hanging fruit,” Benjamini said. “And then we started hitting the wall.”
In 2008, as progress on non-lattice percolation slowed, Schramm died at age 46 in a fall while hiking. “We lost a genius, Oded Schramm, to a tragic accident,” Benjamini said. “And then we needed to wait for some new geniuses to come.”
About a decade ago, the field began to accelerate again. But proving supercritical sharpness remained difficult.
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