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 {±1} 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 2mathopen: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 4ℤ produces a combinatorial cancellation of connection patterns, while only the neutral charge sector survives in the scaling limit.

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BibTeX RIS

BibTeX

@misc{chen2026multipoint,
  title = {{Multi-point functions of a full-plane two-state fuzzy $4$-Potts model}},
  author = {Hong-Bin Chen and Jiaming Xia},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2605.31374},
  eprint = {2605.31374},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2605.31374},
}

RIS

TY  - UNPB
TI  - Multi-point functions of a full-plane two-state fuzzy $4$-Potts model
AU  - Hong-Bin Chen
AU  - Jiaming Xia
PY  - 2026
DA  - 2026/09/20
PB  - arXiv
DO  - 10.48550/arxiv.2605.31374
UR  - https://arxiv.org/abs/2605.31374
AB  - We study the full-plane two-state fuzzy $4$-Potts model, obtained by assigning independent balanced {±1} 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 2mathopen: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 4ℤ produces a combinatorial cancellation of connection patterns, while only the neutral charge sector survives in the scaling limit.
ER  -

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