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arXiv · 2108.04975

Dimers, networks, and cluster integrable systems

Anton Izosimov
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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.

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Anton Izosimov. 2021-08-11. Dimers, networks, and cluster integrable systems. https://arxiv.org/abs/2108.04975

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