In supercritical percolation on infinite transitive graphs, the probability that the origin belongs to a finite cluster of size at least n decays exponentially — but exponentially in what? Not simply in n, but in the isoperimetric function of the graph. If the graph's isoperimetric profile grows linearly, the decay is exponential in n. If the graph is more expansion-rich, the decay is even faster. The sharpness of the supercritical phase is governed by the geometry of the underlying graph, not by the percolation model itself.
This result unifies a family of previously separate analyses. For lattices, hyperbolic spaces, and nonamenable graphs, the same theorem applies — the specific geometry enters only through the isoperimetric function, a single quantity that encodes how efficiently surfaces enclose volume. The percolation transition inherits its sharpness from the ambient geometry, as though the graph's expansion properties dictate how decisively the system commits to its infinite cluster.
The structural claim: phase transitions do not live in the dynamics alone. The sharpness with which a system transitions between phases is borrowed from the geometry of the space it occupies. The container shapes the crisis.
(arXiv:2603.03257)