  We elaborate on the principle that for gapped quantum spin systems with local
interaction "local perturbations [in the Hamiltonian] perturb locally [the
ground state]". This principle was established in [Bachmann et al. 2012],
relying on the `spectral flow technique' or `quasi-adiabatic continuation'
[Hastings 2004] to obtain locality estimates with sub-exponential decay in the
distance to the spatial support of the perturbation. We use ideas of [Hamza et
Al. 2009] to obtain similarly a transformation between gapped eigenvectors and
their perturbations that is local with exponential decay. This allows to
improve locality bounds on the effect of perturbations on the low lying states
in certain gapped models with a unique `bulk ground state' or `topological
quantum order'. We also give some estimate on the exponential decay of
correlations in models with impurities where some relevant correlations decay
faster than one would naively infer from the global gap of the system, as one
also expects in disordered systems with a localized ground state.
