I want to know whether the following version of GKS inequality with boundary condition for Ising model hold or not. Consider Ising model on $\mathbb{Z}^d$ and $\varnothing \neq A\subset \Lambda_1 \subset \Lambda_2 \subset \mathbb{Z}^d$. Let $x \in \Lambda_1 \setminus A$ and $\eta_A$ be a fixed configuration on $A$. Is the following true? \begin{equation} \langle \sigma_x \rangle_{\Lambda_1}^{\eta_A} \leq \langle \sigma_x \rangle_{\Lambda_2}^{\eta_A}, \qquad (\star) \end{equation} where $\langle \cdot \rangle_{\Lambda}^{\eta}$ is the expectation w.r.t to Gibbs measure on $\Lambda$ with boundary condition $\eta$.

I know that the original GKS inequality works with free boundary condition, i.e. $A=\varnothing$, and implies that for all $B \subset \Lambda_1$ we have $\langle \sigma_B \rangle_{\Lambda_1} \leq \langle \sigma_B \rangle_{\Lambda_2}$, with $\sigma_B=\prod_{y\in B} \sigma_y$.

So, can we prove or disprove the inequality $(\star)$?

Thank you.

I want to know whether the following version of GKS inequality with boundary condition for Ising model hold or not. Consider Ising model on $\mathbb{Z}^d$ and $\varnothing \neq A\subset \Lambda_1 \subset \Lambda_2 \subset \mathbb{Z}^d$. Let $x \in \Lambda_1 \setminus A$ and $\eta_A$ be a fixed configuration on $A$. Is the following true? \begin{equation} \langle \sigma_x \rangle_{\Lambda_1}^{\eta_A} \leq \langle \sigma_x \rangle_{\Lambda_2}^{\eta_A}, \qquad (\star) \end{equation} where $\langle \cdot \rangle_{\Lambda}^{\eta}$ is the expectation w.r.t to Gibbs measure on $\Lambda$ with boundary condition $\eta$.

I know that the original GKS inequality works with free boundary condition, i.e. $A=\varnothing$, and implies that for all $B \subset \Lambda_1$ we have $\langle \sigma_B \rangle_{\Lambda_1} \leq \langle \sigma_B \rangle_{\Lambda_2}$, with $\sigma_B=\prod_{y\in B} \sigma_y$.

So, can we prove or disprove the inequality $(\star)$?

Thank you.

Update after the comment by Prof. Yvan Velenik.

As shown by Yvan, the inequality $(\star)$ is not correct in general setting.

I am working with a particular setting as follows. Consider a connected graph $G$ containing several components: $G=A\cup B \cup C$, where $V(A)\cap V(B)=\varnothing$, there is only one edge, say $\{x,y\}$ between $A$ and $B$ with $y \in A$ and $x \in B$; and $C=D \cup E$ with $D\cap A = \varnothing$, $E\cap B=\varnothing$ and $B$ is connected to $D$ and $E$ is connected to both $A$ and $D$. The question is still to check

$$\langle \sigma_x \rangle^{\eta_A}_{B} \leq \langle \sigma_x \rangle^{\eta_A}_G ? \qquad (1)$$

I'am sorry for the complicated construction of $G$. Actually, I'am considering the case $G$ is a random graph locally tree like. I want to truncate the Gibbs measure on $G$ to the measure on a ball (say $B$) around the vertex $x$. So we can expect $A$ and $B$ are very near to trees (there are only a few cycles in $A\cup B$). Assume $|G|=n$ and $|A\cup B|$ is more ore less $(\log n)^3$.

If the inequality is not true, can we expect an approximation like $$\langle \sigma_x \rangle^{\eta_A}_{B} \leq \langle \sigma_x \rangle^{\eta_A}_G + o_n(1)? \qquad (2)$$

where $o_n(1)$ depends only on $n$.

Thank you.