Let $\mathbb Z^2$ denote the two-dimensional integer lattice with norm of $i=(i_1,i_2)$ given by $\|i\|=|i_1|+|i_2|$.

For each $x\in\mathbb Z^2$, we assign a uniform random variable, $\sigma_x$ taking values on the set $\{-1,1\}$.

Fix $\omega\in\{-1,0,1\}^{\mathbb{Z}^2}$ and $\beta>0$. For each finite $\Lambda\subset\mathbb{Z}^d$, define a probability measure on the sigma algebra generated by the cylinder sets of $\{-1,1\}^{\mathbb{Z}^2}$, such that for each $\sigma\in\{-1,1\}^{\mathbb{Z}^2}$ the probability of this configuration is given by $$ \mu_{\Lambda}^{\beta,\omega}(\sigma)= \left\{ \begin{array}{rl} \frac{\exp(-\beta H_{\Lambda}^{\omega}(\sigma))}{Z_{\Lambda}^{\omega}},&\text{if}\ \ \sigma_i=\omega_i\ \forall i\in\Lambda^c;\\ \\ \\ 0,& \text{otherwise}, \end{array} \right. $$ where $$ H_{\Lambda}^{\omega}(\sigma)=-\sum_{i,j\in\Lambda}J_{ij}\sigma_i\sigma_j-\sum_{i\in\Lambda, j\in\Lambda^c}J_{ij}\sigma_i\omega_j $$ with $J_{ij}\equiv J(\|i-j\|)\geq 0$ and $J_{ij}=0$ if $\|i-j\|\geq R$, for some positive $R$ and $Z_{\Lambda}^{\omega}$ is a normalizing constant so that $\mu_{\Lambda}^{\beta,\omega}$ is a probability measure.

**Question 1: ** If $\Lambda_n\uparrow\mathbb{Z}^2$ and $\omega_i=0$ for all $i\in\mathbb{Z}^2$, sounds reasonable that any accumulation point of the sequence $\mu_{\Lambda_n}^{\beta,\omega}$, in the weak* topology, is translation invariant. Is this true for any finite $R$ ?

**Question 2:** Suppose $R$ finite and bigger than one, keeping the setting of Question 1 but $\omega_i=1$ (or $\omega_i=-1$) for all $i\in\mathbb{Z}^2$ is the weak* limit
$$w-\lim_{n\to\infty} \mu_{\Lambda_n}^{\beta,\omega}$$

translational invariant ?

**Question 3: ** For finite $R$ bigger than one is it true the Aizenman-Higuchi Theorem

$$w-\lim_{n\to\infty} \mu_{\Lambda_n}^{\beta,\omega}\in [\mu^{\beta,+},\mu^{\beta,-}] ?$$