  We characterize the carrier density profile of the ground state of graphene
in the presence of particle-particle interaction and random charged impurity
for zero gate voltage. We provide detailed analysis on the resulting spatially
inhomogeneous electron gas taking into account the particle-particle
interaction and the remote coulomb disorder on an equal footing within the
Thomas-Fermi-Dirac theory. We present some general features of the carrier
density probability measure of the graphene sheet. We also show that, when
viewed as a random surface, the resulting electron-hole puddles at zero
chemical potential show peculiar self-similar statistical properties. Although
the disorder potential is chosen to be Gaussian, we show that the charge field
is non-Gaussian with unusual Kondev relations which can be regarded as a new
class of two-dimensional (2D) random-field surfaces.
