Local dimer dynamics in higher dimensions

  • Ivailo Hartarsky

    TU Wien, Vienna, Austria
  • Lyuben Lichev

    Université Jean Monnet, Saint-Etienne, France; Institute of Science and Technology, Klosterneuburg, Austria
  • Fabio Lucio Toninelli

    TU Wien, Vienna, Austria
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Abstract

We consider local dynamics of the dimer model (perfect matchings) on hypercubic boxes . These consist of successively switching the dimers along alternating cycles of prescribed (small) lengths. We study the connectivity properties of the dimer configuration space equipped with these transitions. Answering a question of Freire, Klivans, Milet, and Saldanha, we show that in three dimensions any configuration admits an alternating cycle of length at most 6. We further establish that any configuration on features order alternating cycles of length at most . We also prove that the dynamics of dimer configurations on the unit hypercube of dimension is ergodic when switching alternating cycles of length at most . Finally, in the planar but non-bipartite case, we show that parallelogram-shaped boxes in the triangular lattice are ergodic for switching alternating cycles of lengths 4 and 6 only, thus improving a result of Kenyon and Rémila, which also uses 8-cycles. None of our proofs make reference to height functions.

Cite this article

Ivailo Hartarsky, Lyuben Lichev, Fabio Lucio Toninelli, Local dimer dynamics in higher dimensions. Ann. Inst. Henri Poincaré Comb. Phys. Interact. (2024), published online first

DOI 10.4171/AIHPD/200