Continuum nonsimple loops and 2D critical percolation

Research output: Contribution to journalArticle

Abstract

Substantial progress has been made in recent years on the 2D critical percolation scaling limit and its conformal invariance properties. In particular, chordal SLE 6 (the Stochastic Loewner Evolution with parameter κ=6) was, in the work of Schramm and of Smirnov, identified as the scaling limit of the critical percolation "exploration process." In this paper we use that and other results to construct what we argue is the full scaling limit of the collection of all closed contours surrounding the critical percolation clusters on the 2D triangular lattice. This random process or gas of continuum nonsimple loops in {\Bbb R} 2 is constructed inductively by repeated use of chordal SLE 6. These loops do not cross but do touch each other - indeed, any two loops are connected by a finite "path" of touching loops.

Original languageEnglish (US)
Pages (from-to)157-173
Number of pages17
JournalJournal of Statistical Physics
Volume116
Issue number1-4
DOIs
StatePublished - Aug 2004

Keywords

  • SLE
  • conformal invariance
  • continuum loops
  • nonsimple loops
  • percolation
  • scaling limit
  • triangular lattice

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics

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