▎ 摘　　要

We characterize the carrier density profile of the ground state of graphene in the presence of particle-particle interaction and random charged impurity in zero gate voltage. We provide detailed analysis on the resulting spatially inhomogeneous electron gas, taking into account the particle-particle interaction and the remoteCoulomb 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 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 random-field surfaces. Using Schramm-Loewner (SLE) evolution, we numerically demonstrate that the ungated graphene has conformal invariance and the random zero-charge density contours are SLEk with k = 1.8 +/- 0.2, consistent with c = -3 conformal field theory.