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Cross-ratio dynamics and the dimer cluster integrable system

Abstract : Cross-ratio dynamics, allowing to construct 2D discrete conformal maps from 1D initial data, is a well-known discrete integrable system in discrete differential geometry. We relate it to the dimer integrable system from statistical mechanics by identifying its invariant Poisson structure and integrals of motion recently found by Arnold et al. to the Goncharov-Kenyon counterparts for the dimer model on a specific class of graphs. This solves the open question of finding a cluster algebra structure describing cross-ratio dynamics. The main tool relating geometry to the dimer model is the definition of triple crossing diagram maps associated to bipartite graphs on the cylinder. In passing we write the bivariate polynomial defining the dimer spectral curve for arbitrary bipartite graphs on the torus as the characteristic polynomial of a one-parameter family of matrices, a result which may be of independent interest.
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https://hal.archives-ouvertes.fr/hal-03364744
Contributor : Sanjay Ramassamy Connect in order to contact the contributor
Submitted on : Monday, October 4, 2021 - 9:53:40 PM
Last modification on : Sunday, June 26, 2022 - 3:14:47 AM
Long-term archiving on: : Wednesday, January 5, 2022 - 7:18:24 PM

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  • HAL Id : hal-03364744, version 1

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Niklas Affolter, Terrence George, Sanjay Ramassamy. Cross-ratio dynamics and the dimer cluster integrable system. 2021. ⟨hal-03364744⟩

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