We first consider nested CLE4 in a simply-connected domain and compute some exact renormalised probabilities: the probability that two points belong to the same CLE4 gasket and the probability that two points belong to the outermost CLE4 gasket. The resulting conformally covariant formulas have a non-trivial modular dependence, expressed explicitly in terms of Jacobi theta functions. We are guided by the conformal field theory of the Ashkin-Teller (AT) model, but our proofs are purely probabilistic using Brownian loop soups, gauge-twisted Gaussian free fields (GFFs) and the level line geometry of the 2D continuum GFF. More generally, we also calculate renormalised probabilities that two points belong to CLE4 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; they also suggest a CLE4-based continuum FK representation of the AT single-spin fields along a whole segment of the critical line.
This Is joint work with Juhan Aru (EPFL).