, Volume 139, Issue 3-4, pp 473-519,
Open Access This content is freely available online to anyone, anywhere at any time.

Critical percolation exploration path and SLE 6: a proof of convergence

Abstract

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.

Research of Federico Camia was partially supported by a Marie Curie Intra-European Fellowship under contract MEIF-CT-2003-500740 and by a Veni grant of the Dutch Organization for Scientific Research (NWO).
Research of Charles M.Newman was partially supported by the US NSF under grants DMS-01-04278 and DMS-06-06696.