Take the 2-minute tour ×  Consider the first neighbors Ising
model in $\mathbb Z^2$, with the Hamiltonian in the finite
volume $\Lambda\subset\mathbb{Z}^2$ given by $$ H_{\Lambda}(\sig
ma|\omega)=-J\sum_{i,j\in\Lambda\atop{\|i-
j\|=1}}\sigma_i\sigma_j-J\sum_{i\in\Lambda,
j\in\Lambda^c\atop{\|i-j\|=1}}\sigma_i\omega_j $$ 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 $$ \mu\notin \left\{w-\lim_{\Lambda\uparrow\mathbb{Z}^2}\
\ \mu_{\Lambda}^{\beta,\omega}:\omega\in\{-1,1\}^{\mathbb{Z}^2}
\right\} \ \ ? $$  share|improve this question add comment  1
Answer  up vote 3 down vote accepted  I don't think so. Just
consider Dobrushin boundary conditions (positive spins at
vertices wit