PEPS is a tensor network that plays the same role in two dimensional lattice as MPS 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".