  Graphene is convenient material for nanomechanichal applications since
high-frequency oscillations are easily accessible. In this Article, we consider
graphene on a rough substrate attached to imperfections at random locations. We
explore the statistics of low-lying phonon modes, which exert most influence on
the conductivity of graphene. We find that the nearest neighbor spacings of low
lying eigenfrequencies have the Wigner-Dyson probability distribution after
averaging over the random configurations of disorder. Due to interaction of
electrons with the oscillations of the membrane, an electron can be transfered
to higher or lower energies, which is a manifestation of the phonon-assisted
Tien-Gordon effect. The Tien-Gordon effect suppresses the conductivity of
graphene. In the regime of low Fermi energies and small sizes of the sample an
increase of conductivity is observed which we refer to Klein tunneling and
electron-hole pair creation. Eventually, when the increase of the transmission
becomes too prominent, the pair creation changes the ground state of the
system, signalizing the limit of applicability of the single-particle Dirac
equation used in this paper.
