This paper investigates the nature of ripples in graphene, a two-dimensional (2D) carbon crystal. The study uses atomistic Monte Carlo simulations based on a precise many-body interatomic potential for carbon. The results show that ripples spontaneously appear due to thermal fluctuations, with a size distribution peaked around 70 Å, consistent with experimental observations. This contradicts the current understanding of the stability of flexible membranes, which predicts a power-law scaling of fluctuations at long wavelengths. The unexpected result is attributed to the multiplicity of chemical bonding in carbon. The study also finds that the bending rigidity of graphene increases with temperature, contrary to expectations. This is related to fluctuations in bond length, which signal a partial change from conjugated to single/double bonds, leading to deviations from planarity. The simulations reveal that the typical height of fluctuations scales with the sample size as $ L^{\zeta} $, with $ \zeta = 1 - \eta/2 $, where $ \eta $ is the anomalous rigidity exponent. Despite this, the fluctuations remain anomalously large and can exceed the interatomic distance for large samples. The paper also discusses the existence of long-range crystallographic order in membranes, which can be destroyed by topological defects. However, in graphene, the energy of these defects is too high to be relevant. The results show that the spatial distribution of ripples cannot be described by the general theory of flexible membranes. Instead, the correlation function of the normals shows a maximum at a preferred length scale of about 70 Å, which is also observable in real space images. The findings suggest that the unique properties of carbon bonding make graphene different from a generic 2D crystal. The results have implications for understanding the stability and electronic transport properties of graphene. The study highlights the importance of considering the specific chemical bonding and interatomic interactions in two-dimensional materials.This paper investigates the nature of ripples in graphene, a two-dimensional (2D) carbon crystal. The study uses atomistic Monte Carlo simulations based on a precise many-body interatomic potential for carbon. The results show that ripples spontaneously appear due to thermal fluctuations, with a size distribution peaked around 70 Å, consistent with experimental observations. This contradicts the current understanding of the stability of flexible membranes, which predicts a power-law scaling of fluctuations at long wavelengths. The unexpected result is attributed to the multiplicity of chemical bonding in carbon. The study also finds that the bending rigidity of graphene increases with temperature, contrary to expectations. This is related to fluctuations in bond length, which signal a partial change from conjugated to single/double bonds, leading to deviations from planarity. The simulations reveal that the typical height of fluctuations scales with the sample size as $ L^{\zeta} $, with $ \zeta = 1 - \eta/2 $, where $ \eta $ is the anomalous rigidity exponent. Despite this, the fluctuations remain anomalously large and can exceed the interatomic distance for large samples. The paper also discusses the existence of long-range crystallographic order in membranes, which can be destroyed by topological defects. However, in graphene, the energy of these defects is too high to be relevant. The results show that the spatial distribution of ripples cannot be described by the general theory of flexible membranes. Instead, the correlation function of the normals shows a maximum at a preferred length scale of about 70 Å, which is also observable in real space images. The findings suggest that the unique properties of carbon bonding make graphene different from a generic 2D crystal. The results have implications for understanding the stability and electronic transport properties of graphene. The study highlights the importance of considering the specific chemical bonding and interatomic interactions in two-dimensional materials.