I started to read some articles on graphene and almost all say that graphene was discovered late because physicists thought it would be unstable. Despite this, I didn't found a clear explanation of why graphene is indeed stable.
Why does graphene exist, despite the Mermin-Wagner theorem?
The answer lies in the fact that, in graphene, there is an effective long range interaction mediated by the inverse biharmonic operator (which in 2D goes as $x^2\ln(x)$ and is extremely long-ranged) coupling the gaussian curvature at any two points on the sheet. Due to this, any static ripples or thermally produced dynamic ripples interact at arbitrary distances and allow for the existence of a flat phase (ordering of the normals). I have written an answer here , which basically explains this point. David Nelson and Peliti did quite a bit of work on this in the 1980s, in the context of polymerized or tethered membranes (membranes with crystalline order).
