  We analyze a criterion which guarantees that the ground states of certain
many body systems are stable under perturbations. Specifically, we consider
PEPS, which are believed to provide an efficient description, based on local
tensors, for the low energy physics arising from local interactions. In order
to assess stability in the framework of PEPS, one thus needs to understand how
physically allowed perturbations of the local tensor affect the properties of
the global state. In this paper, we show that a restricted version of the Local
Topological Quantum Order (LTQO) condition provides a checkable criterion which
allows to assess the stability of local properties of PEPS under physical
perturbations. We moreover show that LTQO itself is stable under perturbations
which preserve the spectral gap, leading to nontrivial examples of PEPS which
possess LTQO and are thus stable under arbitrary perturbations.
