We witness many phase transitions in everyday life (eg. ice melting to water). The mathematical approach to these phenomena revolves around the percolation model: given a graph, call each vertex open with probability p independently of the others and look at the subgraph induced by open vertices. Benjamini and Schramm conjectured in 1996 that, at p=1/2, on any planar graph, either there is no infinite connected components or infinitely many.
We prove a stronger version of this conjecture and use this to establish fractal macroscopic behaviour in the loop O(n) model. The latter includes a random discrete Lipschitz surface as a particular case.
Joint work with Matan Harel and Nathan Zelesko.