We consider a quantum spin system in any dimension which has a spectral gap above its ground state energy. We assume that the ground state is frustration-free, locally disordered and does not exhibit long range entanglement. We prove that arbitrary perturbations at the boundary of a large domain do not affect ground state expectation values inside the domain. The proof relies on an isoperimetric inequality for the ground state energy. This is joint work with Wojciech De Roeck, Brecht Donvil, and Martin Fraas.
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