### 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.

Original language | English (US) |
---|---|

Pages (from-to) | 473-519 |

Number of pages | 47 |

Journal | Probability Theory and Related Fields |

Volume | 139 |

Issue number | 3-4 |

DOIs | |

State | Published - Nov 2007 |

### Fingerprint

### Keywords

- Conformal invariance
- Continuum scaling limit
- Critical behavior
- Percolation
- SLE
- Triangular lattice

### ASJC Scopus subject areas

- Mathematics(all)
- Statistics and Probability
- Analysis

### Cite this

**Critical percolation exploration path and SLE6 : A proof of convergence.** / Camia, Federico; Newman, Charles.

Research output: Contribution to journal › Article

*Probability Theory and Related Fields*, vol. 139, no. 3-4, pp. 473-519. https://doi.org/10.1007/s00440-006-0049-7

}

TY - JOUR

T1 - Critical percolation exploration path and SLE6

T2 - A proof of convergence

AU - Camia, Federico

AU - Newman, Charles

PY - 2007/11

Y1 - 2007/11

N2 - 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.

AB - 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.

KW - Conformal invariance

KW - Continuum scaling limit

KW - Critical behavior

KW - Percolation

KW - SLE

KW - Triangular lattice

UR - http://www.scopus.com/inward/record.url?scp=34547143925&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=34547143925&partnerID=8YFLogxK

U2 - 10.1007/s00440-006-0049-7

DO - 10.1007/s00440-006-0049-7

M3 - Article

AN - SCOPUS:34547143925

VL - 139

SP - 473

EP - 519

JO - Probability Theory and Related Fields

JF - Probability Theory and Related Fields

SN - 0178-8051

IS - 3-4

ER -