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In this paper the mono-layer graphene at the charge neutrality point isconsidered whithin Thomas-Fermi-Dirac theory, treating inhomogeneous externalpotentials and electron-electron interactions on equal footing. We present somegeneral considerations concerning the probability measure of the ground statecharge density. The system shows degrees of self-similarity. By analyzing theground 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 hasconformal invariance and the random zero-charge density contours areSLE$_{\kappa}$ with $\kappa=1.8\pm 0.2$.

Conformal symmetry of electron-hole puddles in ungated graphene