Convergence of the two-dimensional random walk loop-soup clusters to CLE

Abstract

We consider the random walk loop-soup of subcritical intensity parameter on the discrete half-plane $\mathtt{H}:=\mathbb{Z}\times \mathbb{N}$. We look at the clusters of discrete loops and show that the scaling limit of the outer boundaries of outermost clusters is a CLE${}_{\kappa }$ conformal loop ensemble.