  We show that a generalized Dirac structure survives beyond the linear regime
of the low-energy dispersion relations of graphene. A generalized uncertainty
principle of the kind compatible with specific quantum gravity scenarios with a
fundamental minimal length (here graphene lattice spacing) and Lorentz
violation (here the particle/hole asymmetry, the trigonal warping, etc.) is
naturally obtained. We then show that the corresponding emergent field theory
is a table-top realization of such scenarios, by explicitly computing the third
order Hamiltonian, and giving the general recipe for any order. Remarkably, our
results imply that going beyond the low-energy approximation does not spoil the
well-known correspondence with analogue massless quantum electrodynamics
phenomena (as usually believed), but rather it is a way to obtain experimental
signatures of quantum-gravity-like corrections to such phenomena.
