Work

Local dimer dynamics in higher dimensions

Year: 2024

Type: article

Abstract: We consider local dynamics of the dimer model (perfect matchings) on hypercubic boxes [n]^{d} . 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 [n]^{d} features order n^{d-2} alternating cycles of length at most 4d-2 . We also prove that the dynamics of dimer configurations on the unit hypercube of dimension d is ergodic when switching alternating cycles of length at most 4d-4 . 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.

Source: Annales de l’Institut Henri Poincaré D Combinatorics Physics and their Interactions

DOI: 10.4171/aihpd/200

Authors: Ivailo Hartarsky, Lyuben Lichev, Fabio Lucio Toninelli

Institutions: TU Wien, Institute of Science and Technology Austria, Université Jean Monnet

Language: English

Topic: Stochastic processes and statistical mechanics

Subfield: Mathematical Physics

Field: Mathematics

Domain: Physical Sciences

Open Access status: diamond

Funder: Austrian Science Fund

Award: G8769036332