Random walk loop soups and conformal loop ensembles

TL;DR

This paper proves that clusters of large loops in the random walk loop soup converge to Brownian loop soup clusters, establishing a connection to conformal loop ensembles and their boundaries in the continuum limit.

Contribution

It extends previous results by showing the convergence of loop clusters and their outer boundaries from lattice models to conformally invariant continuum objects.

Findings

Clusters of large lattice loops converge to Brownian loop soup clusters.

Outer boundaries of these clusters converge to conformal loop ensembles (CLE).

Results bridge discrete loop models and continuum conformal invariants.

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.