Dimers, networks, and cluster integrable systems

Research output: Contribution to journalArticlepeer-review

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.

Original languageEnglish (US)
Pages (from-to)861-880
Number of pages20
JournalGeometric and Functional Analysis
Volume32
Issue number4
DOIs
StatePublished - Aug 2022

ASJC Scopus subject areas

  • Analysis
  • Geometry and Topology

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