The conformally invariant metric on CLE$_4$ III: uniqueness

Abstract

This paper is the third and final article in a series of papers constructing the canonical conformally invariant metric on the set of loops of the conformal loop ensemble (CLE) with critical parameter $\kappa=4$. The previous two articles construct, as a subsequential limit of the renormalized graph metric on the loops of CLE$_\kappa$ as $\kappa \downarrow 4$, a conformally invariant, local metric on the loops of a CLE$_4$ whose metric ball growth from the domain boundary coincides with the uniform exploration of Werner and Wu. In this paper, we establish that this metric is uniquely characterized by its properties, as are its geodesics, and that it is a measurable function of the CLE$_4$. In particular, we show that the renormalized CLE$_\kappa$ graph metric converges as $\kappa \downarrow 4$ without passing to a subsequence. A key step in the proof is to show that the metric is determined by the geodesics from each loop to the domain boundary, which are in turn determined by the uniform exploration; this representation will have important applications in future work.