Renormalised two-point functions of CLE$_4$ gaskets

We consider nested CLE$_4$ in a simply-connected domain and compute the following renormalised probabilities: the probability that two points belong to the same CLE$_4$ gasket and the probability that two points belong to the outermost CLE$_4$ gasket. While the integrability is rooted in the conformal field theory of the Ashkin-Teller (AT) model, we provide a purely probabilistic calculation via Brownian loop soups and the geometry of the 2D continuum Gaussian free field. More generally, we also calculate renormalised probabilities that two points belong to CLE$_4$ gaskets sampled in alternation with certain two-valued sets of the Gaussian free field. These quantities correspond to the two-point function of the conjectured scaling limit of the AT single spins on the critical line. At the decoupling point, our results recover the Ising model correlations and suggest a CLE$_4$-based FK representation of the AT spin model.