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 \[{\{ 0,1\} ^\mathbb{Z}} \] consisting of all \[\sigma % MathType!MTEF!2!1!+- % feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbWexLMBbXgBd9gzLbvyNv2CaeHbl7mZLdGeaGqiVu0Je9sqqr % pepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs % 0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaai % aabeqaamaabaabauaakeaaqaaaaaaaaaWdbiabeo8aZbaa!4104! \]in \[{\{ 0,1\} ^\mathbb{Z}} % MathType!MTEF!2!1!+- % feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbWexLMBbXgBd9gzLbvyNv2CaeHbl7mZLdGeaGqiVu0Je9sqqr % pepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs % 0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaai % aabeqaamaabaabauaakeaaqaaaaaaaaaWdbiaacUhacaaIWaGaaiil % aiaaigdacaGG9bWdamaaCaaaleqabaWdbiablssiIcaaaaa!452A! \] 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 \[{v_0},...,{v_k} % MathType!MTEF!2!1!+- % feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbWexLMBbXgBd9gzLbvyNv2CaeHbl7mZLdGeaGqiVu0Je9sqqr % pepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs % 0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaai % aabeqaamaabaabauaakeaaqaaaaaaaaaWdbiaadAhapaWaaSbaaSqa % a8qacaaIWaaapaqabaGcpeGaaiilaiaac6cacaGGUaGaaiOlaiaacY % cacaWG2bWdamaaBaaaleaapeGaam4AaaWdaeqaaaaa!4725! \]. 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.
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