First-order sentences in random groups II: $\forall\exists$-sentences

We prove that a random group, in Gromov’s density model with d<1/16 , satisfies with overwhelming probability a universal-existential first-order sentence \sigma (in the language of groups) if and only if \sigma is true in a nonabelian free group. It is remarked that one can also obtain the optimal result, that is, replace d<1/16 by d<1/2 .