A single vacancy in an otherwise clean flake of graphene introduces a topological defect. It punches a new state at zero energy into the Dirac cone (zero mode). A finite concentration of such vacancies produces a narrow impurity band that determines the low energy/frequency response of the flake. A peculiarity of this band is that its properties are very sensitive to details of the impurity placings: All impurities in one sublattice imply a density of states (DoS) vanishing in the zero energy limit, while impurities distributed equally over both sublattices (compensated) exhibit a diverging DoS. We present a numerical study focussing mainly on the DoS at very low energies. We resolve the Dyson-singularity with compensated disorder and investigate the amplitude statistics (multifractality) of the associated wavefunctions. Finally, we present transport calculations addressing the possibility of Anderson-localization on percolation clusters preempting the classical localization transition.
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