Zero-one law for random subgraphs of some distance graphs with vertices in $\mathbb Z^n$

The zero-one law for the model of random distance graphs with vertices in $\mathbb Z^n$ is studied. Sufficient conditions for a sequence of random distance graphs to obey the zero-one law are derived, as well as conditions under which it contains a subsequence obeying the zero-one law.
Bibliography: 20 titles.