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.
This is a preview of subscription content, log in via an institution to check access.
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
References
N. Affolter, M. Glick, P. Pylyavskyy, and S. Ramassamy. Vector-relation configurations and plabic graphs. Sém. Lothar. Combin., 84B (2020)
D. Cimasoni and N. Reshetikhin. Dimers on surface graphs and spin structures. I. Comm. Math. Phys., 275(1) (2007), 187–208
V. Fock and A. Goncharov. Cluster ensembles, quantization and the dilogarithm. Ann. Sci. Éc. Norm. Supér., 42(6) (2009), 865–930
V.V. Fock and A. Marshakov. Loop groups, clusters, dimers and integrable systems. In Geometry and quantization of moduli spaces, Springer, New York, (2016), pp. 1–65
S. Fomin and A. Zelevinsky. Cluster algebras I: Foundations. J. Am. Math. Soc., 15(2) (2002), 497–529
M. Gekhtman, M. Shapiro, S. Tabachnikov, and A. Vainshtein. Integrable cluster dynamics of directed networks and pentagram maps. Adv. Math., 300 (2016), 390–450
M. Gekhtman, M. Shapiro, and A. Vainshtein. Cluster algebras and Poisson geometry. Mosc. Math. J., 3(3) (2003), 899–934
M. Gekhtman, M. Shapiro, and A. Vainshtein. Poisson geometry of directed networks in a disk. Selecta Math., 15(1) (2009), 61–103
M. Gekhtman, M. Shapiro, and A. Vainshtein. Poisson geometry of directed networks in an annulus. J. Eur. Math. Soc., 14(2) (2012), 541–570
M. Glick. The pentagram map and Y-patterns. Adv. Math., 227(2) (2011), 1019–1045
A.B. Goncharov and R. Kenyon. Dimers and cluster integrable systems. Ann. Sci. Éc. Norm. Supér., 46(5) (2013), 747–813
P. Kasteleyn. The statistics of dimers on a lattice: I. The number of dimer arrangements on a quadratic lattice. Physica, 27(12) (1961), 1209–1225
R. Kenyon and A. Okounkov. Planar dimers and Harnack curves. Duke Math. J., 131(3) (2006), 499–524
R. Kenyon, A. Okounkov, and S. Sheffield. Dimers and amoebae. Ann. Math., 163 (2006), 1019–1056
A. Postnikov. Total positivity, Grassmannians, and networks. arXiv:math/0609764, 2006
R. Schwartz. The pentagram map. Exp. Math., 1(1) (1992), 71–81
K. Talaska. Determinants of weighted path matrices. arXiv:1202.3128, 2012
Acknowledgements
The author is grateful to Michael Gekhtman and Pavlo Pylyavskyy for fruitful conversations and useful remarks. This work was supported by NSF grant DMS-2008021.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Izosimov, A. Dimers, networks, and cluster integrable systems. Geom. Funct. Anal. 32, 861–880 (2022). https://doi.org/10.1007/s00039-022-00605-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00039-022-00605-8