I have a hard time understanding how to reason with these questions
Let $G \subset \mathbb{Z}^2$ be a bounded connected domain with $-$ boundary conditions. Consider the Ising model on $G$ with parameter $\beta >0$. There exist $x$ and $y$ in $G$ such that $\mathbb{E}[\sigma_x]\mathbb{E}[\sigma_y] \leq 0$. True or False?
Let $G \subset \mathbb{Z}^2$ be a bounded connected domain with $+$ boundary conditions. Consider the Ising model in the low temperature expansion (LTE) on $G$ with parameter $\beta >0$. For any distinct $x$ and $y$ in $G$, $\mathbb{E}[\sigma_x\sigma_y] $ is strictly smaller than
$$ \mathbb{P} [ x \text{ and } y \text{ are not separated by any loop in the LTE}] - \mathbb{P} [ x \text{ and } y \text{ are separated by at least one loop in the LTE}] $$ True or false?
I think the way to solve these questions is with parity of loops which are surrounding $x$. Indeed with boundary condition $+$, in the Ising model in LTE I know (but I don't understand very well why) that the value of a spin at a given site $x$ is the parity of the number of loops $N$ which are surrounding the site $x$ in the low-temperature representation, i.e. $$\mathbb{E}_{+}[\sigma_x] = \mathbb{P}[ N \text{ even }] - \mathbb{P}[N \text{ odd}] $$ For the first zero ideas. But for question 2 my intuition say to me that is something like that (but actually I don't know):
$$ \mathbb{E}[\sigma_x \sigma_y] = \mathbb{P}[ x,y \text{ separated by even number of loops }] - \mathbb{P}[x,y \text{ separated by an odd number of loops}] $$ $$ =2\mathbb{P}[x,y \text{ separated by an even number of loops}] - 1 $$ $$\geq \mathbb{P}[x,y \text{ separated by zero loops}] - (1-\mathbb{P}[x,y \text{ separated by zero loops}])$$ $$ = \mathbb{P}[x,y \text{ separated by zero loops}] - \mathbb{P}[x,y \text{ separated by at least one loops}] $$