Consider the two-dimensional Ashkin-Teller model on the square lattice $\mathbb{Z}^2$ with Hamiltonian:
$$ H = - \sum_{\langle i,j \rangle} \left[ K \sigma_i \sigma_j + K \tau_i \tau_j + k \sigma_i \sigma_j \tau_i \tau_j \right], $$
where $\sigma_i, \tau_i \in \{+1, -1\}$ are Ising spin variables at each lattice site $i$, and the sum runs over nearest-neighbor pairs $\langle i,j \rangle$.
Prove that for the antiferromagnetic regime, specifically when $k < 0$ and $|k| > 2|K|$, there is no percolation of any spin configuration in the thermodynamic limit.
I was considering a transformation of the spin variables that maps the antiferromagnetic model to a ferromagnetic model with different coupling parameters. Alsi I was exploring the use of correlation inequalities, such as the FKG inequality (if applicable) or other relevant inequalities, to bound the probability of percolation.
