The Full Scaling Limit of Two-Dimensional Critical Percolation

Details The Full Scaling Limit of Two-Dimensional Critical Percolation The Full Scaling Limit of Two-Dimensional Critical Percolation

2005-04-02

arxiv.org/abs/math/0504036v1

Mathematics

Probability

67 pages, 22 figures

Scientific paper

We use SLE(6) paths to construct a process of continuum nonsimple loops in the plane and prove that this process coincides with the full continuum scaling limit of 2D critical site percolation on the triangular lattice -- that is, the scaling limit of the set of all interfaces between different clusters. Some properties of the loop process, including conformal invariance, are also proved. In the main body of the paper these results are proved while assuming, as argued by Schramm and Smirnov, that the percolation exploration path converges in distribution to the trace of chordal SLE(6). Then, in a lengthy appendix, a detailed proof is provided for this convergence to SLE(6), which itself relies on Smirnov's result that crossing probabilities converge to Cardy's formula.

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