Critical percolation exploration path and SLE6: a proof of convergence

By Federico Camia, Charles M. Newman. 2006

167 citations recorded by the source corpus.

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 SLE6. 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 SLE6 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.

Subfields: Condensed Matter Physics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/abc4646d335a7478fa1a957693c80d1742b05cd3

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In Condensed Matter Physics

In Mathematical Physics

In Statistics and Probability

Citations in the corpus

Reversibility of chordal SLE (2008, background): “So far it has been proved that the chordal SLE6 is the scaling limit of the explorer line of the site percolation on the triangular lattice with half open and half closed boundary conditions ([13] and [2]); the chordal SLE8 is the scaling limit of UST Peano curve with half free and half wired bounda”
Two-Dimensional Critical Percolation: The Full Scaling Limit (2006, background): “However, a detailed proof of statement (S), based on Smirnov’s theorem about the convergence of crossing probabilities to Cardy’s formula [36], is now the topic of a separate paper [12].”