  In this paper the mono-layer graphene at the charge neutrality point is
considered whithin Thomas-Fermi-Dirac theory, treating inhomogeneous external
potentials and electron-electron interactions on equal footing. We present some
general considerations concerning the probability measure of the ground state
charge density. The system shows degrees of self-similarity. By analyzing the
ground state carrier density profile, we show that although it is not Gaussian,
the critical exponents are consistent with Kondev hyper-scaling relations.
Using Schramm-Loewner (SLE) evolution we show that the ungated graphene has
conformal invariance and the random zero-charge density contours are
SLE$_{\kappa}$ with $\kappa=1.8\pm 0.2$.
