Translation invariant realizability problem on the d-dimensional lattice: an explicit construction

We consider a particular instance of the truncated realizability problem on the d-dimensional lattice. Namely, given two functions ρ_1(i) and ρ_2(i; j) non-negative and symmetric on ℤ^d, we ask whether they are the first two correlation functions of a translation invariant point process. We provide an explicit construction of such a realizing process for any d ≥ 2 when the radial distribution has a specific form. We also derive from this construction a lower bound for the maximal realizable density and compare it with the already known lower bounds.