  We study the effects of long-range electrostatic interactions on the thermal
fluctuations of free-standing crystalline membranes exhibiting spontaneous
electric polarization directed at each point along the local normal to the
surface. We show that the leading effect of dipole-dipole interactions in the
long-wavelength limit consists in renormalizations of the bending rigidity and
the elastic coefficients. A completely different result was obtained in the
case of scalar two-point interactions decaying as $R^{-3}$, where $R$ is the
distance. In the latter case, which was addressed in previous theoretical
research, the energy of long-wavelength bending fluctuations is controlled by
power-law interactions and it scales with the wavevector $q$ as $q^{3}$,
leading to a modified large-distance behaviour of correlation functions. By
contrast, in the case of dipole interactions, the $q^{3}$ dependence of the
bending energy vanishes. Non-local terms generated by the expansion of the
electrostatic energy are suppressed in the limit of small wavevectors. This
suggests that the universal scaling behaviour of elastic membranes holds even
in presence of dipole interactions. At the same time, the shift of the Lam\'e
coefficients and the bending rigidity induced by electrostatic interactions can
be quantitatively important for two-dimensional materials with a permanent
out-of-plane polarization.
