In material science research, I am developing an algorithm to solve an infinite combinatorial optimization problem which I believe is the most natural problem when the system size goes to infinity.

What's the current status in mathematics/computer science of this infinite combinatorial optimization problem?

Given an interaction range $k+1$, and interaction parameters $v_0,v_1,\ldots,v_k$, $$ \inf_{\sigma \in S} \left( \lim_{N \rightarrow \infty} \frac{1}{N} \sum_{ i \in \{-N,\ldots,N\} } \left( v_0 \sigma_i + \sum_{ j \in \{1,\ldots,k\} } v_j \sigma_i \sigma_{i + j} \right) \right) $$ where the minimization is over the subset $S$ of $\{0,1\}^{\Bbb Z}$ consisting of all $\sigma \in \{0,1\}^{\Bbb Z}$ for which the above limit exists.

The physical intuition is that each atom can be in 2 spin states, 0 or 1, and they have interactions defined by $v_0$,...,$v_k$. We want to find the ground state of the system.

I currently have an algorithm and code that could compute on a personal computer for k<~25 and I am planning for a publication on it. Further, if you are also interested in this kind of problem (2D, 3D or even higher dimension) and want to collaborate. Please don't hesitate to contact me:) Thank you.