We study the properties of the set of marginal distributions of infinitetranslation-invariant systems in the 2D square lattice. In cases where thelocal variables can only take a small number $d$ of possible values, wecompletely solve the marginal or membership problem for nearest-neighborsdistributions ($d=2,3$) and nearest and next-to-nearest neighbors distributions($d=2$). Remarkably, all these sets form convex polytopes in probability space.This allows us to devise an algorithm to compute the minimum energy per site ofany TI Hamiltonian in these scenarios exactly. We also devise a simplealgorithm to approximate the minimum energy per site up to arbitrary accuracyfor the cases not covered above. For variables of a higher (but finite)dimensionality, we prove two no-go results. To begin, the exact computation ofthe energy per site of arbitrary TI Hamiltonians with only nearest-neighborinteractions is an undecidable problem. In addition, in scenarios with $d\geq2947$, the boundary of the set of nearest-neighbor marginal distributionscontains both flat and smoothly curved surfaces and the set itself is notsemi-algebraic. This implies, in particular, that it cannot be characterizedvia semidefinite programming, even if we allow the input of the program toinclude polynomials of nearest-neighbor probabilities.
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机译：我们研究了二维方格中无限平移不变系统的边际分布集的性质。在局部变量只能取少量$ d $可能值的情况下，我们完全解决了最近邻分布（$ d = 2,3 $）以及最近邻和最近邻分布（$ d = 2 $）。值得注意的是，所有这些集合都在概率空间中形成了凸多边形，这使我们能够设计出一种算法来精确计算这些场景下任何TI哈密顿量的每个位置的最小能量。对于上面未涵盖的情况，我们还设计了一种简单算法，可以将每个站点的最小能量近似为任意精度。对于较高（但有限）维的变量，我们证明了两个不合格的结果。首先，仅具有最近邻相互作用的任意TI哈密顿量的每位能量的精确计算是一个不确定的问题。此外，在具有$ d \ geq2947 $的情况下，最近邻边际分布集的边界既包含平坦曲面，也包含平滑曲面，并且集本身不是半代数的。这特别意味着，即使我们允许程序的输入包含最邻近概率的多项式，也无法通过半定值编程对其进行表征。
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