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We characterize the carrier density profile of the ground state of graphenein the presence of particle-particle interaction and random charged impurityfor zero gate voltage. We provide detailed analysis on the resulting spatiallyinhomogeneous electron gas taking into account the particle-particleinteraction and the remote coulomb disorder on an equal footing within theThomas-Fermi-Dirac theory. We present some general features of the carrierdensity probability measure of the graphene sheet. We also show that, whenviewed as a random surface, the resulting electron-hole puddles at zerochemical potential show peculiar self-similar statistical properties. Althoughthe disorder potential is chosen to be Gaussian, we show that the charge fieldis non-Gaussian with unusual Kondev relations which can be regarded as a newclass of two-dimensional (2D) random-field surfaces.

Scale-invariant puddles in Graphene: Geometric properties of electron-hole distribution at the Dirac point