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I have a hard time understanding how to reason with these questions

  1. 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?

  2. 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):

$$ \mathbb{E}[\sigma_x \sigma_y] = \mathbb{P}[ x,y \text{ separated by even number of loops }] - \mathbb{P}[x,y \text{ separated by an odd number of loops}] $$ $$ =2\mathbb{P}[x,y \text{ separated by an even number of loops}] - 1 $$ $$\geq \mathbb{P}[x,y \text{ separated by zero loops}] - (1-\mathbb{P}[x,y \text{ separated by zero loops}])$$ $$ = \mathbb{P}[x,y \text{ separated by zero loops}] - \mathbb{P}[x,y \text{ separated by at least one loops}] $$

3m0o
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    For your first question, do you agree that this is equivalent to having the same inequality for $+$ boundary condition? Now, for two number $A,B$, to have $A \cdot B <0 $, you need $A>0>B$ or $B>0>A$, would this be possible here? A similar idea holds for $AB=0$. – Kernel Jan 20 '23 at 15:05
  • For the second question, I didn't follow your first identity. – Kernel Jan 20 '23 at 15:05
  • I don't see why the same inequality has to hold with $+$ boundary condition. Anyway seem reasonable that $ \mathbb{E}[\sigma_x] < 0 $ for all site $x$ with $-$ boundary condition (hence the results is false). And $ \mathbb{E}[\sigma_x] > 0$ for all site $x$ for $+$ boundary condition. But I don't know how to prove it or I don't see if this is trivial :) for the second question, the first equality was kind of a guess. If for $\mathbb{E}[\sigma_x] = \mathbb{P}[N \text{even}] - \mathbb{P}[N \text{odd}]$ I thought that similar equality has to holds also for $\mathbb{E}[\sigma_x\sigma_y]$ – 3m0o Jan 20 '23 at 16:13

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