Critical Percolation Exploration Path and SLE(6): a Proof of Convergence

Details Critical Percolation Exploration Path and SLE(6): a Proof of Convergence Critical Percolation Exploration Path and SLE(6): a Proof of Convergence

2006-04-22

arxiv.org/abs/math/0604487v2

Mathematics

Probability

45 pages, 14 figures; revised version following the comments of a referee

Scientific paper

It was argued by Schramm and Smirnov that the critical site percolation exploration path on the triangular lattice converges in distribution to the trace of chordal SLE(6). We provide here a detailed proof, which relies on Smirnov's theorem that crossing probabilities have a conformally invariant scaling limit (given by Cardy's formula). The version of convergence to SLE(6) that we prove suffices for the Smirnov-Werner derivation of certain critical percolation crossing exponents and for our analysis of the critical percolation full scaling limit as a process of continuum nonsimple loops.

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