  We consider a q-deformed version of the uniform Gibbs measure on dimers on
the periodized hexagonal lattice (equivalently, on interlacing particle
configurations, if vertical dimers are seen as particles) and show that it is
invariant under a certain irreversible q-Whittaker dynamic. Thereby we provide
a new non-trivial example of driven interacting two-dimensional particle
system, or of (2+1)-dimensional stochastic growth model, with explicit
stationary measure. We emphasize that this measure is far from being a product
Bernoulli measure. These Gibbs measures and dynamics both arose earlier in the
theory of Macdonald processes. The q=0 degeneration of the Gibbs measures
reduce to the usual uniform dimer measures with given tilt, the degeneration of
the dynamics originate in the study of Schur processes and the degeneration of
the results contained herein were recently treated in work of the second
author.
