  The many-body theory of interacting electrons poses an intrinsically
difficult problem that requires simplifying assumptions. For the determination
of electronic screening properties of the Coulomb interaction, the Random Phase
Approximation (RPA) provides such a simplification. Here, we explicitly show
that this approximation is justified for band structures with sizeable band
gaps. This is when the electronic states responsible for the screening are
energetically far away from the Fermi level, which is equivalent to a short
electronic propagation length of these states. The RPA contains exactly those
diagrams in which the classical Coulomb interaction covers all distances,
whereas neglected vertex corrections involve quantum tunneling through the
barrier formed by the band gap. Our analysis of electron-electron interactions
provides a real-space analogy to Migdal's theorem on the smallness of vertex
corrections in electron-phonon problems. An important application is the
increasing use of constrained Random Phase Approximation (cRPA) calculations of
effective interactions. We find that their usage of Kohn-Sham energies already
accounts for the leading local (excitonic) vertex correction in insulators.
