### Abstract

We analyze the geometry of scaling limits of near-critical 2D percolation, i.e., for p = p_{c}+λδ^{1/ν}, with ν = 4/3, as the lattice spacing δ → 0. Our proposed framework extends previous analyses for p = p_{c}, based on SLE_{6}. It combines the continuum nonsimple loop process describing the full scaling limit at criticality with a Poissonian process for marking double (touching) points of that (critical) loop process. The double points are exactly the continuum limits of "macroscopically pivotal" lattice sites and the marked ones are those that actually change state as λ varies. This structure is rich enough to yield a one-parameter family of near-critical loop processes and their associated connectivity probabilities as well as related processes describing, e.g., the scaling limit of 2D minimal spanning trees.

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

Pages (from-to) | 1155-1171 |

Number of pages | 17 |

Journal | Journal of Statistical Physics |

Volume | 125 |

Issue number | 5-6 |

DOIs | |

State | Published - Dec 2006 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Physics and Astronomy(all)
- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Journal of Statistical Physics*,

*125*(5-6), 1155-1171. https://doi.org/10.1007/s10955-005-9014-6

**The scaling limit geometry of near-critical 2D percolation.** / Camia, Federico; Fontes, Luiz Renato G; Newman, Charles.

Research output: Contribution to journal › Article

*Journal of Statistical Physics*, vol. 125, no. 5-6, pp. 1155-1171. https://doi.org/10.1007/s10955-005-9014-6

}

TY - JOUR

T1 - The scaling limit geometry of near-critical 2D percolation

AU - Camia, Federico

AU - Fontes, Luiz Renato G

AU - Newman, Charles

PY - 2006/12

Y1 - 2006/12

N2 - We analyze the geometry of scaling limits of near-critical 2D percolation, i.e., for p = pc+λδ1/ν, with ν = 4/3, as the lattice spacing δ → 0. Our proposed framework extends previous analyses for p = pc, based on SLE6. It combines the continuum nonsimple loop process describing the full scaling limit at criticality with a Poissonian process for marking double (touching) points of that (critical) loop process. The double points are exactly the continuum limits of "macroscopically pivotal" lattice sites and the marked ones are those that actually change state as λ varies. This structure is rich enough to yield a one-parameter family of near-critical loop processes and their associated connectivity probabilities as well as related processes describing, e.g., the scaling limit of 2D minimal spanning trees.

AB - We analyze the geometry of scaling limits of near-critical 2D percolation, i.e., for p = pc+λδ1/ν, with ν = 4/3, as the lattice spacing δ → 0. Our proposed framework extends previous analyses for p = pc, based on SLE6. It combines the continuum nonsimple loop process describing the full scaling limit at criticality with a Poissonian process for marking double (touching) points of that (critical) loop process. The double points are exactly the continuum limits of "macroscopically pivotal" lattice sites and the marked ones are those that actually change state as λ varies. This structure is rich enough to yield a one-parameter family of near-critical loop processes and their associated connectivity probabilities as well as related processes describing, e.g., the scaling limit of 2D minimal spanning trees.

KW - Finite size scaling

KW - Minimal spanning tree

KW - Near-critical

KW - Percolation

KW - Scaling limits

UR - http://www.scopus.com/inward/record.url?scp=33845733049&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33845733049&partnerID=8YFLogxK

U2 - 10.1007/s10955-005-9014-6

DO - 10.1007/s10955-005-9014-6

M3 - Article

VL - 125

SP - 1155

EP - 1171

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

IS - 5-6

ER -