Stationary measure of the driven two-dimensional $q$-Whittaker particle system on the torus

Abstract

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 [7]. The $q=0$ degeneration of the Gibbs measures reduce to the usual uniform dimer measures with given tilt [12], the degeneration of the dynamics originate in the study of Schur processes [5, 6] and the degeneration of the results contained herein were recently treated in [19].