Energy spectrum for charge carriers in graphene with folded deformations or with flexural modes with Gaussian and L\'evy distributed random pseudo-magnetic fields.

The electronic behaviour in graphene under a flexural field with random height displacements, considered as pseudo-magnetic fields, is studied. General folded deformations (not necessarily random) were first studied, giving an expression for the zero energy modes. For Gaussian folded deformations, it is possible to use a Coulomb gauge norm for the fields allowing contact with previous work on the quantum Hall effect with random fields, showing that the density of states has a power law behaviour and that the zero energy modes wavefunctions are multifractal. This hints of an unusual electron velocity distribution. Also, an Aharonov-Bohm pseudo-effect is produced. For more general non-folded general flexural strain, is not possible to use a Coulomb gauge. However, a Random Phase Approximation (RPA) and the scheme of random matrix theory allows to tackle the problem. For Gaussian distributed fields, the spectrum presents an average gap and for some cases, a breaking of the particle-hole symmetry. Finally, for the case of L\'evy distributed fields, nearly flat bands are seen due to strong electron localization.