  In various approaches to quantum gravity continuum spacetime is expected to
emerge from discrete geometries through a phase transition. In group field
theory, various indications for such a transition have recently been found but
a complete understanding of such a phenomenon remains an open issue. In this
work, we investigate the critical behavior of different group field theory
models in the Gaussian approximation. Applying the Ginzburg criterion to
quantify field fluctuations, we find that this approximation breaks down in the
case of three-dimensional Euclidean quantum gravity as described by the
dynamical Boulatov model on the compact group $\text{SU}(2)$. This result is
independent of the peculiar gauge symmetry and specific form of nonlocality of
the model. On the contrary, we find that the Gaussian approximation is valid
for a rank-$1$ GFT on the noncompact sector of fields on
$\text{SL}(2,\mathbb{R})$ related to Lorentzian models. Though a
nonperturbative analysis is needed to settle the question of phase transitions
for compact groups, the results may also indicate the necessity to consider
group field theory on noncompact domains for phase transitions to occur.
