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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.

Two Dimensional Translation-Invariant Probability Distributions: Approximations, Characterizations and No-Go Theorems