  The stability of two-dimensional (2D) layers and membranes is subject of a
long standing theoretical debate. According to the so called Mermin-Wagner
theorem, long wavelength fluctuations destroy the long-range order for 2D
crystals. Similarly, 2D membranes embedded in a 3D space have a tendency to be
crumpled. These dangerous fluctuations can, however, be suppressed by
anharmonic coupling between bending and stretching modes making that a
two-dimensional membrane can exist but should present strong height
fluctuations. The discovery of graphene, the first truly 2D crystal and the
recent experimental observation of ripples in freely hanging graphene makes
these issues especially important. Beside the academic interest, understanding
the mechanisms of stability of graphene is crucial for understanding electronic
transport in this material that is attracting so much interest for its unusual
Dirac spectrum and electronic properties. Here we address the nature of these
height fluctuations by means of straightforward atomistic Monte Carlo
simulations based on a very accurate many-body interatomic potential for
carbon. We find that ripples spontaneously appear due to thermal fluctuations
with a size distribution peaked around 70 \AA which is compatible with
experimental findings (50-100 \AA) but not with the current understanding of
stability of flexible membranes. This unexpected result seems to be due to the
multiplicity of chemical bonding in carbon.
