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

We prove that a random group, in Gromov’s density model with $d<1\mathrm{/}16$ 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.