Percolation on the projective plane

Abstract

Since the projective plane is closed, the natural homological observable of a percolation process is the presence of the essential cycle in H1(RP2; Z2). In the Voroni model at critical phase, pc = .5, this observable has probability q = .5 independent of the metric on RP2. This establishes a single instance (RP2, homological observable) of a very general conjecture about the conformal invariance of percolation due to Aizenman and Langlands, for which there is much moral and numerical evidence but no previously verified instances. On RP2 all metrics are conformally equivalent so the proof of metric independence is precisely what the conjecture would predict. What is very special, is that at pc metric invariance holds in all finite models so passing to the limit is trivial; the probability q is fixed at .5 by a topological symmetry.