### Abstract

Under some general assumptions, we construct the scaling limit of open clusters and their associated counting measures in a class of two dimensional percolation models. Our results apply, in particular, to critical Bernoulli site percolation on the triangular lattice and to the critical FK–Ising model on the square lattice. Fundamental properties of the scaling limit, such as conformal covariance, are explored. As an application to Bernoulli percolation, we obtain the scaling limit of the largest cluster in a bounded domain. We also apply our results to the critical, two-dimensional Ising model, obtaining the existence and uniqueness of the scaling limit of the magnetization field, as well as a geometric representation for the continuum magnetization field which can be seen as a continuum analog of the FK representation of the Ising model.

Original language | English |
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Title of host publication | Sojourns in Probability Theory and Statistical Physics - II |

Subtitle of host publication | Brownian Web and Percolation, A Festschrift for Charles M. Newman |

Editors | Vladas Sidoravicius |

Publisher | Springer |

Pages | 44-89 |

Number of pages | 46 |

Volume | 2 |

ISBN (Electronic) | 9789811502989 |

ISBN (Print) | 9789811502972 |

DOIs | |

Publication status | Published - 2019 |

Event | International Conference on Probability Theory and Statistical Physics, 2016 - Shanghai, China Duration: 25 Mar 2016 → 27 Mar 2016 |

### Publication series

Name | Springer Proceedings in Mathematics and Statistics |
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Volume | 299 |

ISSN (Print) | 2194-1009 |

ISSN (Electronic) | 2194-1017 |

### Conference

Conference | International Conference on Probability Theory and Statistical Physics, 2016 |
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Country | China |

City | Shanghai |

Period | 25/03/16 → 27/03/16 |

### Keywords

- Critical cluster
- Ising model
- Magnetization field
- Percolation
- Scaling limit
- Schramm–Smirnov topology

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## Cite this

*Sojourns in Probability Theory and Statistical Physics - II: Brownian Web and Percolation, A Festschrift for Charles M. Newman*(Vol. 2, pp. 44-89). (Springer Proceedings in Mathematics and Statistics; Vol. 299). Springer. https://doi.org/10.1007/978-981-15-0298-9_2