Critical percolation exploration path and SLE
6: a proof of convergence |
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Authors: | Federico Camia Charles M Newman |
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Institution: | (1) Department of Mathematics, Vrije Universiteit Amsterdam, Amsterdam, The Netherlands;(2) Courant Institute of Mathematical Sciences, New York University, New York, USA |
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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 Charles M.Newman was partially supported by the US NSF under grants DMS-01-04278 and DMS-06-06696. |
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Keywords: | Continuum scaling limit Percolation SLE Critical behavior Triangular lattice Conformal invariance |
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