### Abstract

We postulate the existence of a natural Poissonian marking of the double (touching) points of SLE _{6} and hence of the related continuum nonsimple loop process that describes macroscopic cluster boundaries in 2D critical percolation. We explain how these marked loops should yield continuum versions of near-critical percolation, dynamical percolation, minimal spanning trees and related plane filling curves, and invasion percolation. We showthat this yields for some of the continuum objects a conformal covariance property that generalizes the conformal invariance of critical systems. It is an open problem to rigorously construct the continuum objects and to prove that they are indeed the scaling limits of the corresponding lattice objects.

Original language | English (US) |
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Pages (from-to) | 537-559 |

Number of pages | 23 |

Journal | Bulletin of the Brazilian Mathematical Society |

Volume | 37 |

Issue number | 4 |

DOIs | |

State | Published - Dec 2006 |

### Keywords

- Conformal covariance
- Finite size scaling
- Minimal spanning tree
- Near-critical
- Off-critical
- Percolation
- Scaling limits

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

*Bulletin of the Brazilian Mathematical Society*,

*37*(4), 537-559. https://doi.org/10.1007/s00574-006-0026-x