On August 31, Quanta Magazine ran a headline calling it "Stunning": a paper by five people at ETH Zurich—Diskin, Easo, Radhakrishnan, Sudakov, and Tassion—titled simply *Supercritical sharpness of percolation* (arXiv:2603.03257). Fellow researcher Nachmias said, "I find great joy in this proof. It's stunning." Itai Benjamini repeated twice: "It's a gem."
The game itself. Take an infinite graph where each edge is independently "open" with probability p. Starting from the origin, what is the probability of reaching infinity? Call it θ(p). Intuition suggests a critical point pc: below it, θ(p) = 0 and the world stays disconnected; above it, θ(p) becomes positive and an infinite cluster appears. Sharpness asks whether that "immediately" is really true—whether the phase transition is a steep cliff or a murky slope.
What the paper actually proves. The statement "an infinite cluster exists for every p > pc" is trivial by definition—pc is the infimum where θ turns positive, and monotonicity comes free. The real content is the strength of the other side: for every p > pc, the probability that the origin's cluster is finite but has size at least n is bounded by e^(−cΦ(n)), where Φ is the graph's isoperimetric profile. In plain terms: past the critical point, particles still trapped in finite clusters die off exponentially at the fastest rate the graph's geometry allows. As a bonus, the proof recovers the Grimmett–Marstrand 1990 slab theorem and yields a transience corollary. The key move is running the old "sprinkling" (extra edge-opening) technique in reverse: analyze the sprinkled edges first. Just flipping it around.
A puzzle four decades in the making. In 1986–87, Menshikov in Moscow and Aizenman–Barsky in New Jersey proved sharpness on the standard lattice Z^d. In 1996, Benjamini–Schramm published *Percolation beyond Z^d*, pushing the question to all "uniform-looking" (transitive) graphs—Quanta's thirty-year countdown starts there. The remaining difficulty was carved up by graph class and eliminated piece by piece:
- Nonamenable graphs: solved 2021, Hermon–Hutchcroft (Invent. Math.)
- Amenable, polynomial growth (e.g., lattices): solved 2024, Contreras–Martineau–Tassion (Duke)
- Amenable, superpolynomial growth (certain groups and graphs): solved by this paper—the last piece
A story like a novel. The week before Christmas 2025, at an ETH blackboard: two postdocs (Diskin, Easo), one PhD student (Radhakrishnan), plus Sudakov and Tassion. On the morning of December 17, everyone was convinced; the complete proof was on paper before Christmas. Tassion: "From there, it was in my head day and night." Diskin added: "He barely slept for at least two weeks." The paper hit arXiv on March 3, 2026, with Quanta's coverage on August 31. It has not been submitted to any journal yet—v1 is public, and anyone can dig in.
What remains open: behavior exactly at p = pc—in particular whether θ(pc) = 0 on the three-dimensional lattice, one of the original Benjamini–Schramm conjectures. Sharpness says the cliff is steep; it says nothing about what happens at the cliff's tip.
Why it's worth an afternoon. Percolation is the fruit fly of phase-transition theory: forest fires, power-grid cascades, material conductivity, epidemic thresholds—all share this mathematical skeleton. Extending sharpness from lattices to all uniform graphs means these systems' "critical cliffs" no longer depend on the regularity of space—however strange the geometry, the phase transition is clean. For statistical mechanics, this is a foundational step.
References: Quanta coverage | Paper, arXiv:2603.03257 | Duminil-Copin–Tassion 2016 sharpness framework | Antunić–Veselić 2007