Random walk loop soups and conformal loop ensembles

Tim van de Brug, Federico Camia, Marcin Lis

Research output: Contribution to journalArticle

Abstract

The random walk loop soup is a Poissonian ensemble of lattice loops; it has been extensively studied because of its connections to the discrete Gaussian free field, but was originally introduced by Lawler and Trujillo Ferreras as a discrete version of the Brownian loop soup of Lawler and Werner, a conformally invariant Poissonian ensemble of planar loops with deep connections to conformal loop ensembles (CLEs) and the Schramm–Loewner evolution (SLE). Lawler and Trujillo Ferreras showed that, roughly speaking, in the continuum scaling limit, “large” lattice loops from the random walk loop soup converge to “large” loops from the Brownian loop soup. Their results, however, do not extend to clusters of loops, which are interesting because the connection between Brownian loop soup and CLE goes via cluster boundaries. In this paper, we study the scaling limit of clusters of “large” lattice loops, showing that they converge to Brownian loop soup clusters. In particular, our results imply that the collection of outer boundaries of outermost clusters composed of “large” lattice loops converges to CLE.

Original languageEnglish (US)
Pages (from-to)553-584
Number of pages32
JournalProbability Theory and Related Fields
Volume166
Issue number1-2
DOIs
StatePublished - Oct 1 2016

Keywords

  • Brownian loop soup
  • Conformal loop ensemble
  • Outer boundary
  • Planar Brownian motion
  • Random walk loop soup

ASJC Scopus subject areas

  • Analysis
  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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