Consider the Ferromagnetic Ising Model ($J>0$) on the lattice $\mathbb{Z}^2$ with the Hamiltonian with boundary condition $\omega\in\{-1,1\}$ formally given by $$ H^{\omega}_{\Lambda}(\sigma)=-J\sum_{<i,j>}{\sigma_i\sigma_j} - \sum_{i\in\Lambda} {h_i\sigma_i}, $$ where the first sum is over all unordered pairs of first neighbors in $\Lambda\cup\partial \Lambda$.
Suppose that $h_i=h>0$ if $i\in\Gamma\subset\mathbb{Z}^d$ and $h_i=0$ for $i\in \mathbb{Z}^2\setminus \Gamma$. If $\Gamma$ is a finite set then this model has transition.
I expect that if $\Gamma$ is a subset of $\mathbb{Z}^2$ very sparse, for instance $\Gamma=\mathbb{P}\times\mathbb{P}$, where $\mathbb{P}$ is the set of prime numbers this model also has phase transition. So my question is, there exist an explicit example where $\Gamma$ is an infinite set and this model has phase transition?