Renormalised two-point functions of CLE gaskets
arXiv:2604.15146
Abstract
We first consider nested CLE in a simply-connected domain and compute some exact renormalised probabilities: the probability that two points belong to the same CLE gasket and the probability that two points belong to the outermost CLE 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 CLE 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 CLE-based continuum FK representation of the AT single-spin fields along a whole segment of the critical line.
76 pages, beautiful figures are still missing; revision contains a more explicit calculation of 1pt functions using spectral zeta functions