  We study the distribution of vacuum polarization charge induced by a Coulomb
impurity in massive graphene. By analytically computing the polarization
function, we show that the charge density is distributed in space in a
non-trivial fashion, and on a characteristic length-scale set by the effective
Compton wavelength. The density crosses over from a logarithmic behavior below
this scale, to a power law variation above it. Our results in the continuum
limit are confirmed by explicit diagonalization of the corresponding
tight-binding model on a finite-size lattice. Electron-electron interaction
effects are also discussed.
