### 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) |
---|---|

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 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Bulletin of the Brazilian Mathematical Society*,

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

**Two-dimensional scaling limits via marked nonsimple loops.** / Camia, Federico; Fontes, Luiz Renato G; Newman, Charles.

Research output: Contribution to journal › Article

*Bulletin of the Brazilian Mathematical Society*, vol. 37, no. 4, pp. 537-559. https://doi.org/10.1007/s00574-006-0026-x

}

TY - JOUR

T1 - Two-dimensional scaling limits via marked nonsimple loops

AU - Camia, Federico

AU - Fontes, Luiz Renato G

AU - Newman, Charles

PY - 2006/12

Y1 - 2006/12

N2 - 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.

AB - 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.

KW - Conformal covariance

KW - Finite size scaling

KW - Minimal spanning tree

KW - Near-critical

KW - Off-critical

KW - Percolation

KW - Scaling limits

UR - http://www.scopus.com/inward/record.url?scp=33845800266&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33845800266&partnerID=8YFLogxK

U2 - 10.1007/s00574-006-0026-x

DO - 10.1007/s00574-006-0026-x

M3 - Article

VL - 37

SP - 537

EP - 559

JO - Bulletin of the Brazilian Mathematical Society

JF - Bulletin of the Brazilian Mathematical Society

SN - 1678-7544

IS - 4

ER -