Random walk loop soups and conformal loop ensembles
Artikel i vetenskaplig tidskrift, 2016

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.

Random walk loop soup



Outer boundary





critical percolation

Planar Brownian motion

Brownian loop soup


T. van de Brug

Vrije Universiteit Amsterdam

F. Camia

New York University Abu Dhabi

Vrije Universiteit Amsterdam

Marcin Lis

Göteborgs universitet

Chalmers, Matematiska vetenskaper, Matematisk statistik

Probability Theory and Related Fields

0178-8051 (ISSN) 1432-2064 (eISSN)

Vol. 166 1 553-584





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