Bulletin of the Brazilian Mathematical Society

, Volume 37, Issue 4, pp 537–559 | Cite as

Two-dimensional scaling limits via marked nonsimple loops

  • Federico Camia
  • Luiz Renato G. Fontes
  • Charles M. Newman
Article

Abstract.

We postulate the existence of a natural Poissonian marking of the double (touching) points of SLE6 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.

Keywords:

scaling limits percolation near-critical off-critical minimal spanning tree finite size scaling conformal covariance 

Mathematical subject classification:

Primary: 60K35 82B43 82B27 Secondary: 60G57 60K37 82B24 82B28 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Federico Camia
    • 1
  • Luiz Renato G. Fontes
    • 2
  • Charles M. Newman
    • 3
  1. 1.Department of MathematicsVrije UniversiteitAmsterdamNETHERLANDS
  2. 2.Instituto de Matemática e EstatísticaUniversidade de São PauloSão PauloBRAZIL
  3. 3.Courant Inst. of Mathematical Sciences New York UniversityNew YorkU.S.A.

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