  It has been proved that in gapped ground states of locally-interacting
quantum systems, the effect of local perturbations decays exponentially with
distance. However, in systems with power-law ($1/r^\alpha$) decaying
interactions, no analogous statement has been shown, and there are serious
mathematical obstacles to proving it with existing methods. In this paper we
prove that when $\alpha$ exceeds the spatial dimension $D$, the effect of local
perturbations on local properties a distance $r$ away is upper bounded by a
power law $1/r^{\alpha_1}$ in gapped ground states, provided that the
perturbations do not close the spectral gap. The power-law exponent $\alpha_1$
is tight if $\alpha>2D$ and interactions are two-body, where we have
$\alpha_1=\alpha$. The proof is enabled by a method that avoids the use of
quasiadiabatic continuation and incorporates techniques of complex analysis.
This method also improves bounds on ground state correlation decay, even in
short-range interacting systems. Our work generalizes the fundamental notion
that local perturbations have local effects to power-law interacting systems,
with broad implications for numerical simulations and experiments.
