Consider the lattice $\mathbb Z^2$ and take iid random variables $Y_e$ on all edges $e$ of the graph.

We then define random variables $X_i:=\sum_{e \text{ adjacent to } i}Y_e.$

In other words: For every vertex we define a random variable $X_i$ that is the sum of the $Y_e$ on its neighbouring edges.

I ask: Is there a probability space $(\Omega,\mathcal A, \mathbb P)$ such that this stochastic process $(X_i)$ is ergodic with respect to the standard shift $(T_i)_{i \in \mathbb Z^2}$ where $(T_i f)(z)=f(z-i)$? If the $X_i$ were independent this would be easy as the product measure would do it. But they are only almost independent in the sense that $X_i$ and $X_j$ are independent if $\vert i-j\vert \ge 2$.

I recall for you:

A stochastic process is called ergodic if there is an ergodic family of measure preserving transformations $(T_i)$ such that $X_i(T_j\omega)= X_{i-j}(\omega).$

Now, the shift is a reasonable candidate to be measure preserving and it satisfies also the property $X_i(T_j\omega)= X_{i-j}(\omega)$

Can we also realize it as an ergodic family?

Let me add that the situation does not seem to be that bad by noting that due to independence on edges we have that

$$\frac{\sum_{\vert i \vert \le n} X_i}{n^2} \rightarrow 4\mathbb E(Y_e).$$