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 ndβ2 alternating cycles of length at most 4dβ2. 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
4dβ4. 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 R\'emila, which also uses 8-cycles. None of our proofs make reference to
height functions.Comment: 14 pages, 4 figure