  Proving that the parent Hamiltonian of a Projected Entangled Pair State
(PEPS) is gapped remains an important open problem. We take a step forward in
solving this problem by showing two results: first, we identify an approximate
factorization condition on the boundary state of rectangular subregions that is
sufficient to prove that the parent Hamiltonian of the bulk 2D PEPS has a
constant gap in the thermodynamic limit; second, we then show that Gibbs state
of a local, finite-range Hamiltonian satisfy such condition. The proof applies
to the case of injective and MPO-injective PEPS, employs the martingale method
of nearly commuting projectors, and exploits a result of Araki on the
robustness of one dimensional Gibbs states. Our result provides one of the
first rigorous connections between boundary theories and dynamical properties
in an interacting many body system.
