Multi-point functions of a full-plane two-state fuzzy $4$-Potts model

Abstract

We study the full-plane two-state fuzzy $4$-Potts model, obtained by assigning independent balanced $\{\pm1\}$ spins to the open clusters of a critical $q=4$ random-cluster configuration. Its spin field has the same law as a single-spin projection and as the polarization field of the isotropic Ashkin--Teller model at its Potts point. We prove that, after proper normalization, all even multi-point spin correlation functions converge to explicit conformally covariant Coulomb-gas type neutral charge sums. As a consequence, we prove convergence in law of the rescaled magnetization field and identify its limit in law with $2\mathopen{:}\cos(X/2)\mathclose{:}$ for a full-plane Gaussian free field $X$. The proof combines the Baxter--Kelland--Wu coupling, convergence of the six-vertex height function to the Gaussian free field, and a charge-completion mechanism: an enlarged discrete sum over charge assignments with total charge in $4\mathbb Z$ produces a combinatorial cancellation of connection patterns, while only the neutral charge sector survives in the scaling limit.