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We consider a particular instance of the truncated realizability problem onthe $d-$dimensional lattice. Namely, given two functions $\rho_1({\bf i})$ and$\rho_2({\bf i},{\bf j})$ non-negative and symmetric on $\mathbb{Z}^d$, we askwhether they are the first two correlation functions of a translation invariantpoint process. We provide an explicit construction of such a realizing processfor any $d\geq 2$ when the radial distribution has a specific form. We alsoderive from this construction a lower bound for the maximal realizable densityand compare it with the already known lower bounds.

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