  We calculate Root Mean Square (RMS) deviations from equilibrium for atoms in
a two dimensional crystal with local (e.g. covalent) bonding between close
neighbors. Large scale Monte Carlo calculations are in good agreement with
analytical results obtained in the harmonic approximation. When motion is
restricted to the plane, we find a slow (logarithmic) increase in fluctuations
of the atoms about their equilibrium positions as the crystals are made larger
and larger. We take into account fluctuations perpendicular to the lattice
plane, manifest as undulating ripples, by examining dual layer systems with
coupling between the layers to impart local rigidly (i.e. as in sheets of
graphene made stiff by their finite thickness). Surprisingly, we find a rapid
divergence with increasing system size in the vertical mean square deviations,
independent of the strength of the interplanar coupling. We consider an
attractive coupling to a flat substrate, finding that even a weak attraction
significantly limits the amplitude and average wavelength of the ripples. We
verify our results are generic by examining a variety of distinct geometries,
obtaining the same phenomena in each case.
