In the paper arXiv:quant-ph/0601075 the authors introduce an interesting correspondence between Projected Entangled Pair States (PEPS) and classical statistical physics. Basically, for any locally-interacting classical statistical system with partition function $$Z=\sum_{\{\sigma\}} e^{-\beta E[\{\sigma\}]},$$ where $\{\sigma\}$ collectively label the classical spin configuration, we can associate a PEPS state $$|\psi\rangle=\sum_{\{\sigma\}} e^{-\frac{\beta}{2} E[\{\sigma\}]}|\{\sigma\}\rangle,$$ such that $Z=\langle\psi|\psi\rangle$, and there is an exact mapping between thermal expectation values and quantum expectation values, e.g. $$\langle\sigma_j\rangle_{\beta}=\langle\psi|\hat{\sigma}^z_j|\psi\rangle/\langle\psi|\psi\rangle.$$ [Note that $|\psi\rangle$ can be easily shown to be a PEPS since $e^{-\frac{\beta}{2} E[\{\sigma\}]}$ is a product of local Boltzmann weights.]
So every locally-interacting classical statistical system can be mapped to a PEPS. I want to know under what condition is the reverse direction true, i.e. which family of PEPS can be mapped to a classical statistical system in the above way? It is clear that if the local PEPS tensor takes only non-negative values, then $\langle\psi|\psi\rangle$ can be interpreted as a classical partition function with strictly local interactions. But this is not a necessary condition-- even if the local tensor takes negative values sometimes, it is possible that it becomes completely positive after doing a suitable gauge transformation.
[one motivation to ask this question is that, as we know, local properties of a general PEPS can not be calculated as easily as MPS in 1D; yet if we know an exact correspondence to a statistical model, its properties can be more easily understood from that classical model, e.g. by doing Monte-Carlo simulations]
