  Long-wavelength Mermin-Wagner fluctuations prevent the existence of
translational long-range order, in two-dimensional systems at finite
temperature. Their dynamical signature, which is the divergence of the
vibrational amplitude with the system size, also affects disordered solids and
washes out the transient solid-like response generally exhibited by liquids
cooled below their melting temperature. Through a combined numerical and
experimental investigation, here we show that long-wavelength fluctuations are
also relevant at high temperature, where the liquid dynamics does not reveal a
transient solid-like response. In this regime, they induce an unusual but
ubiquitous decoupling between long-time diffusion coefficient $D$ and
structural relaxation time $\tau$, where $D\propto \tau^{-\kappa}$, with
$\kappa > 1$. Long-wavelength fluctuations have a negligible influence on the
relaxation dynamics only at extremely high temperatures, in molecular liquids,
or extremely low densities, in colloidal systems.
