Exponential decay of correlation in PEPS PEPS (Projected Entangled Pair State) is a tensor network that plays the same role in two dimensional lattice as MPS (Matrix Product State) plays in one dimensional spin chain. A good introduction can be found at : http://arxiv.org/abs/1306.2164
For MPS, a very simple criterion exists to ensure an exponential decay of correlation using the technique of transfer matrix. Are there also some criteria (which are easy to verify) that ensure that a given PEPS has an exponential decay of correlation? 
One obvious criterion would be that the parent hamiltonian of the given PEPS be gapped. But I guess this itself is hard to check and hence does not fall under  "easy to verify".
 A: There are no easy to verify criteria which will work in general.  One way to see it is by noting that for every classical StatMech model, we can define a PEPS with the same correlation functions (for which the tensors can be easily constructed from the StatMech model), see http://arxiv.org/abs/quant-ph/0601075.  On the other hand, for StatMech models it is generally a hard problem to determine e.g. the exact point of the phase transition, which in turn relates to the behavior of the correlation functions.
Of course, for restricted cases you can have such criteria.  For instance, if you can derive a parent Hamiltonian for which you can prove that there exists a gap (such as in an environment of an RG fixed point, see e.g. Appendix E of http://arxiv.org/abs/1010.3732), then you can use exponential clustering and you are good.  There will certainly exist a number of other such scenarios where you can prove the existence of exponentially decaying correlations, but without further specification of what kind of PEPS you are looking for, this is not really answerable.
