Background:
Let $\mathbb Z^d$ denote the $d$-dimensional integer lattice with norm $|x|=\sum_i|x_i|$. For each $x\in\mathbb Z^d$ we associate a spin variable, $\sigma_x$ taking values on the set $\{\exp(2\pi j/N);1\leq j\leq N\}$$\{\exp(2\pi i j/N);1\leq j\leq N\}$ with uniform a priori distribution.
Consider the formal Hamiltonian given on the lattice $\mathbb Z^d$ by
$$
H_{ \Lambda }(\{\sigma\}) = -\sum_{\langle x,y\rangle} J_{xy} \vec\sigma_{x} \vec\sigma_{y}
$$
where $J_{xy}$ are nonnegative constants and $\vec\sigma_x \vec\sigma_y$ is inner product in $\mathbb R^2$.
The sum is taken over all pair of first neighbors $\langle x,y\rangle$ means that $|x-y|=1$. The Partition function on a finite
$\Lambda\subset \mathbb Z^d$ is given by
$$
Z_\Lambda = \int\exp\Big( \beta \sum_{\langle x,y\rangle \in \Lambda }J_{xy} \vec\sigma_x \vec\sigma_y\Big)\;d\sigma
$$
where the integral is taken over all sites of $\Lambda$. The two point correlations are given by
$$
\langle \vec\sigma_x \vec\sigma_y \rangle_{\Lambda}=
Z_{\Lambda}^{-1}\int\vec\sigma_x \vec\sigma_y\exp\Big( \beta \sum_{\langle x,y\rangle \in \Lambda }J_{xy} \vec\sigma_x \vec\sigma_y\Big) d\sigma
$$
Question:
For which values of $N$ is known that the Lieb-Simon Inequality is true or false ?
Lieb-Simon Inequality $$ \langle \vec\sigma_x \vec\sigma_y \rangle_{\Lambda} \leq \sum_{b \in \partial B} \bigl<\vec\sigma_x \vec\sigma_b \bigr>_B \bigl<\vec\sigma_b \vec\sigma_y \bigr>_\Lambda, $$ where $B\subset\Lambda\subset\mathbb Z^d$ are finite, $x,y\in\Lambda$, $\partial B=\{z\in B; d(z,B^c)=1\}$ and $\partial B$ separates $x$ and $y$ ($i.e.$ any path from $x$ to $y$ must intercept $\partial B$).