Lengthy logarithms and graphene's Debye-Waller factor

In an infinite, two-dimensional crystal, long wavelength thermal phonons create a divergence in the mean-square displacement $u_p^2$ of atoms from their ideal lattice positions, which has led some to infer that the existence of graphene might depend on the stabilizing influence of ripples in the third dimension. Using the Debye model to approximate graphene's phonon band structure, we calculate $u_p^2$ and the resulting Debye-Waller suppression of high order peaks in graphene's electron diffraction pattern. We find that at room temperature in a 10~$\mu$m sample $\sqrt{u_p^2}$ is less than 5\% of the carbon-carbon bond length, well below the Lindemann melting threshold. Our TEM measurements of the Debye-Waller factor in suspended, exfoliated graphene agree with the calculation. Finite size effects are sufficient to explain graphene's evident stability at room temperature. Surprisingly, in the case of graphene even $6\times 10^{23}$ carbon atoms, representing a sheet 126~m on a side, are not enough to approximate an infinitely large crystal.