Random walk loop soups and conformal loop ensembles

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.

30 pages, 7 figures, to appear in Probab. Theory Related Fields

References in corpus (3)

• A Guide to Stochastic Loewner Evolution and its Applications
• Two-Dimensional Critical Percolation: The Full Scaling Limit
• Conformal Correlation Functions in the Brownian Loop Soup

Cited by in corpus (6)

• The first passage sets of the 2D Gaussian free field: convergence and isomorphisms
• Scalar conformal primary fields in the Brownian loop soup
• Coexistence, enhancements and short loops in random walk loop soups
• Crossing exponent in the Brownian loop soup
• Percolation for two-dimensional excursion clouds and the discrete Gaussian free field
• The expectation value of the number of loops and the left-passage probability in the double-dimer model