Dimers, networks, and cluster integrable systems

We prove that the class of cluster integrable systems constructed by Goncharov and Kenyon out dimer model on a torus coincides with one defined Gekhtman, Shapiro, Tabachnikov, Vainshtein using Postnikov’s perfect networks. To end we express characteristic polynomial network’s boundary measurement matrix in terms partition function associated bipartite graph. Our main tool is flat geometry. Namely, show if network drawn such way edges are Euclidian geodesics, then angles between endow graph canonical fractional Kasteleyn orientation. That orientation used to relate measurements.