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We analyze the spontaneous rippling of graphene membranes as function of thecoupling between lattice deformations and electrons.  We numerically study a model of an elastic membrane coupled to Diracfermions. We identify a phase transition from a flat to a rippled configurationof the membrane when increasing the coupling and propose a scaling procedurethat allows us to effectively reach arbitrary large system sizes. We find thatthe critical value of the coupling rapidly decays as the system increases itssize, in agreement with the experimental observation of an unavoidable stablerippled state for suspended graphene membranes. This decay turns out to becontrolled by a power law with a critical exponent $\sim 1/2$.

Large-scale critical behavior of the rippling phase transition for graphene membranes