Bulletin of the Brazilian Mathematical Society

, Volume 37, Issue 4, pp 537–559

Two-dimensional scaling limits via marked nonsimple loops

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

DOI: 10.1007/s00574-006-0026-x

Cite this article as:
Camia, F., Fontes, L.R.G. & Newman, C.M. Bull Braz Math Soc, New Series (2006) 37: 537. doi:10.1007/s00574-006-0026-x


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.


scaling limitspercolationnear-criticaloff-criticalminimal spanning treefinite size scalingconformal covariance

Mathematical subject classification:

Primary: 60K3582B4382B27Secondary: 60G5760K3782B2482B28

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.