How complex are random graphs in first order logic?

It is not hard to write a first order formula which is true for a given graph G but is false for any graph not isomorphic to G. The smallest number D(G) of nested quantifiers in such a formula can serve as a measure for the "first order complexity" of G. Here, this parameter is studied for random graphs. We determine it asymptotically when the edge probability p is constant; in fact, D(G) is of order log n then. For very sparse graphs its magnitude is θ(n). On the other hand, for certain (carefully chosen) values of p the parameter D(G) can drop down to the very slow growing function log* n, the inverse of the TOWER-function. The general picture, however, is still a mystery.