This study reports the observation of electron-hole puddles in graphene using a scanning single electron transistor (SET). The researchers image the carrier density landscape near the neutrality point, revealing electron-hole puddles as predicted by theory. They also determine the local density of states in graphene, finding that the kinetic contribution to the density of states quantitatively explains the measured signal. The chemical potential and its derivative with respect to density, known as the inverse compressibility, provide insight into Coulomb interactions in graphene. The study shows that the inverse compressibility of graphene has an unusual density dependence with a singularity at the neutrality point. The results suggest that exchange and correlation effects are either weak or cancel out. The local compressibility measurements are performed using a SET, which measures the local electrostatic potential with high sensitivity and spatial resolution. The inverse compressibility is determined by monitoring the change in the local electrostatic potential when modulating the carrier density. The study maps the density fluctuations in the graphene sheet using the inverse compressibility as a function of back-gate voltage or density. The density variations are observed on a scale of approximately 150 nm, limited by the spatial resolution of the SET. The results show that the density fluctuations are on the order of $ \Delta n_{2D,B=0T} = \pm3.9 \cdot 10^{10} \, cm^{-2} $. The study also compares the density fluctuations extracted from inverse compressibility measurements with those extracted from surface potential measurements, finding striking quantitative agreement. The results show that the intrinsic disorder length scale in graphene is approximately 30 nm. At high carrier densities, the high compressibility of graphene smoothes out the disorder landscape, but as one approaches the neutrality point, screening becomes poor and the intrinsic disorder length becomes relevant. The study concludes that the intrinsic disorder length scale in graphene is approximately 30 nm.This study reports the observation of electron-hole puddles in graphene using a scanning single electron transistor (SET). The researchers image the carrier density landscape near the neutrality point, revealing electron-hole puddles as predicted by theory. They also determine the local density of states in graphene, finding that the kinetic contribution to the density of states quantitatively explains the measured signal. The chemical potential and its derivative with respect to density, known as the inverse compressibility, provide insight into Coulomb interactions in graphene. The study shows that the inverse compressibility of graphene has an unusual density dependence with a singularity at the neutrality point. The results suggest that exchange and correlation effects are either weak or cancel out. The local compressibility measurements are performed using a SET, which measures the local electrostatic potential with high sensitivity and spatial resolution. The inverse compressibility is determined by monitoring the change in the local electrostatic potential when modulating the carrier density. The study maps the density fluctuations in the graphene sheet using the inverse compressibility as a function of back-gate voltage or density. The density variations are observed on a scale of approximately 150 nm, limited by the spatial resolution of the SET. The results show that the density fluctuations are on the order of $ \Delta n_{2D,B=0T} = \pm3.9 \cdot 10^{10} \, cm^{-2} $. The study also compares the density fluctuations extracted from inverse compressibility measurements with those extracted from surface potential measurements, finding striking quantitative agreement. The results show that the intrinsic disorder length scale in graphene is approximately 30 nm. At high carrier densities, the high compressibility of graphene smoothes out the disorder landscape, but as one approaches the neutrality point, screening becomes poor and the intrinsic disorder length becomes relevant. The study concludes that the intrinsic disorder length scale in graphene is approximately 30 nm.