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Percolation on the projective plane. (English) Zbl 0902.60085

Summary: 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 Voronoi 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.

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B43 Percolation
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