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Critical percolation: the expected number of clusters in a rectangle

Abstract

We show that for critical site percolation on the triangular lattice two new observables have conformally invariant scaling limits. In particular the expected number of clusters separating two pairs of points converges to an explicit conformal invariant. Our proof is independent of earlier results and SLE techniques, and might provide a new approach to establishing conformal invariance of percolation.

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Correspondence to Clément Hongler.

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Hongler, C., Smirnov, S. Critical percolation: the expected number of clusters in a rectangle. Probab. Theory Relat. Fields 151, 735–756 (2011). https://doi.org/10.1007/s00440-010-0313-8

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Mathematics Subject Classification (2000)

  • Primary 60K35
  • Secondary 30C35
  • 81T40
  • 82B43