Zero-one laws for random dictance graphs

We consider zero-one laws for some models of random distance graphs. A random graph is said to follow the zero-one law if for any first-order property the probability that the random graph posseses this property tends either to 0 or to 1. We give conditions under which random dictance graphs follow the zero-one law and conditions under which there exists a subsequence of random distance graphs following the zero-one law. We also consider zero-one laws for formulae with bounded quantifier depth and some other weakened versions of zero-one laws.