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The random walk loop soup is a Poissonian ensemble of lattice loops; it hasbeen extensively studied because of its connections to the discrete Gaussianfree field, but was originally introduced by Lawler and Trujillo Ferreras as adiscrete version of the Brownian loop soup of Lawler and Werner, a conformallyinvariant Poissonian ensemble of planar loops with deep connections toconformal loop ensembles (CLEs) and the Schramm-Loewner evolution (SLE).  Lawler and Trujillo Ferreras showed that, roughly speaking, in the continuumscaling limit, ``large'' lattice loops from the random walk loop soup convergeto ``large'' loops from the Brownian loop soup. Their results, however, do notextend to clusters of loops, which are interesting because the connectionbetween Brownian loop soup and CLE goes via cluster boundaries. In this paper,we study the scaling limit of clusters of ``large'' lattice loops, showing thatthey converge to Brownian loop soup clusters. In particular, our results implythat the collection of outer boundaries of outermost clusters composed of``large'' lattice loops converges to CLE.

Random walk loop soups and conformal loop ensembles