A conformally invariant metric on CLE(4)

Werner and Wu introduced a conformally invariant way of exploring the loops in a CLE$_4$ $\Gamma$ in a simply connected domain. Using the relationship between CLE$_4$ and the Gaussian free field, we show that the dynamics of this exploration process are a deterministic function of the CLE$_4$ loops, and we use this fact to construct a conformally invariant metric on $\Gamma$ for which a ball of radius $t$ coincides with the set of loops explored up to time $t$ by the exploration process. It is conjectured that this metric space is related in the $\epsilon \to 0$ limit to the contact graph metric on CLE$_{4+\epsilon}$ as well as the contact graph metric on $\epsilon$-neighborhoods of CLE$_4$ loops.