Two-dimensional scaling limits via marked nonsimple loops

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

Research output: Contribution to journalArticle

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 languageEnglish (US)
Pages (from-to)537-559
Number of pages23
JournalBulletin of the Brazilian Mathematical Society
Volume37
Issue number4
DOIs
StatePublished - Dec 2006

Keywords

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

ASJC Scopus subject areas

  • Mathematics(all)

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