Consider the first neighbors Ising model in $\mathbb Z^2$, with the Hamiltonian in the finite volume $\Lambda\subset\mathbb{Z}^2$ given by 
$$
H\_{\Lambda}(\sigma|\omega)=-J\sum\_{i,j\in\Lambda\atop{\|i-j\|=1}}\sigma\_i\sigma\_j-J\sum\_{i\in\Lambda, j\in\Lambda^c\atop{\|i-j\|=1}}\sigma\_i\omega\_j
$$
where $\omega\in\{-1,1\}^{\mathbb{Z}^2}$ is a boundary condition. 


By the Aizenman-Higuchi Theorem for any $\beta>0$, we have that closed convex hull of the weak limits of Gibbs measures in finite volume is the convex set 
$
[\mu^{\beta,+},\mu^{\beta,-}].
$


**Question:**  Is there any $\beta>\beta\_c$ and $\lambda\in(0,1)$ such that 
$$
\mu=\lambda\mu^{\beta,+}+(1-\lambda)\mu^{\beta,-}
$$
and  

$$
\mu\notin \left\{w-\lim\_{\Lambda\uparrow\mathbb{Z}^2}\ \ \mu\_{\Lambda}^{\beta,\omega}:\omega\in\{-1,1\}^{\mathbb{Z}^2} \right\} \ \ ?
$$



Is there an example of Gibbs measure that is not a weak limit of finite volume Gibbs measure ?
I don't think so. Just consider Dobrushin boundary conditions (positive spins at vertices with nonnegative second coordinate, negative elsewhere), and a box of the form
$$
\Lambda\_n=\{-n,\ldots,n\}\times\{-n-[a\sqrt{n}],\ldots,n-[a\sqrt{n}]\}.
$$
Then the mixture you'll get in the limit will have $\lambda$ equal to the probability that the open contour passes below $0$, which should go continuously from $1$ to $0$ as $a$ goes from $-\infty$ to $+\infty$ (it is known that the interface converges weakly to a Brownian bridge under diffusive scaling).


Note that this is very much a two-dimensional phenomenon. In 3d, at low enough temperature, I strongly doubt that you can find boundary conditions such that the limiting state is a nontrivial mixture of, say, Dobrushin states. (Of course, it is always true that you can reach *extremal* states in this way.)


