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. 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.
Highlights
Irreversible Markovian dynamics on two-dimensional dimers or interlaced particle configurations are closely related to driven interacting particle systems as well as random surface growth models in (2 + 1)-dimensions
We consider a q-deformed version of the uniform Gibbs measure on dimers on the periodized hexagonal lattice and show that it is invariant under a certain irreversible q-Whittaker dynamic
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 were recently treated in [19]
Summary
Irreversible Markovian dynamics on two-dimensional dimers or interlaced particle configurations (see Figure 3) are closely related to driven interacting particle systems as well as random surface growth models in (2 + 1)-dimensions. A crucial ingredient used in this extension was the fact that, for q = 0, the infinite-volume Gibbs states are known and have an explicit determinantal structure and GFF-like height fluctuations [12] All of this is missing in the q ∈ (0, 1) case, so at present, the extension of our result to the infinite lattice and the proof of its KPZ class behavior is an open problem. Borodin-Bufetov [4] considered another deformation of the Borodin-Ferrari particle system (q = 0 case of our dynamics) and they showed that the invariant measures are given by 6-vertex Gibbs measures This deformation is different from the one we consider here (ours originates from q-Whittaker processes [7] and the other from vertex models [3]).
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