Noetherian criteria for dimer algebras

Let $A$ be a nondegenerate dimer (or ghor) algebra on a torus, and let $Z$ be its center. Using cyclic contractions, we show the following are equivalent: $A$ is noetherian; $Z$ is noetherian; $A$ is a noncommutative crepant resolution; each arrow of $A$ is contained in a perfect matching whose complement supports a simple module; and the vertex corner rings $e_iAe_i$ are pairwise isomorphic.
Comment: 20 pages. Replaces part of arXiv:1412.1750, with new results. Fixed problem of figures not displaying properly in the PDF of v2