Dimers, networks, and cluster integrable systems

TL;DR

This paper proves the equivalence of two classes of cluster integrable systems derived from dimers and perfect networks on a torus, using flat geometry and boundary measurement matrices.

Contribution

It establishes the equivalence between Goncharov-Kenyon and Gekhtman-Shapiro-Tabachnikov-Vainshtein cluster integrable systems through geometric and combinatorial methods.

Findings

Characteristic polynomial expressed via dimer partition function

Edges as Euclidean geodesics induce a fractional Kasteleyn orientation

Boundary measurements relate to the partition function

Abstract

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