Percolation on the projective plane

Since the projective plane is closed, the natural homological observable of a percolation process is the presence of the essential cycle in $H_1(RP^2; Z_2)$. In the Voroni model at critical phase, $p_c=.5$, this observable has probability $q=.5$ independent of the metric on $RP^2$. This establishes a single instance ($RP^2$, 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 $RP^2$ 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 $p_c$ 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.

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Published 1 January 1997