Open Access
2016 Stationary measure of the driven two-dimensional $q$-Whittaker particle system on the torus
Ivan Corwin, Fabio Lucio Toninelli
Electron. Commun. Probab. 21: 1-12 (2016). DOI: 10.1214/16-ECP4624


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].


Download Citation

Ivan Corwin. Fabio Lucio Toninelli. "Stationary measure of the driven two-dimensional $q$-Whittaker particle system on the torus." Electron. Commun. Probab. 21 1 - 12, 2016.


Received: 12 October 2015; Accepted: 11 May 2016; Published: 2016
First available in Project Euclid: 26 May 2016

zbMATH: 1343.82028
MathSciNet: MR3510252
Digital Object Identifier: 10.1214/16-ECP4624

Primary: 60J10 , 60K35 , 82C20 , 82C24

Keywords: $q$-Whittaker particle system , Interacting particle system , Interface growth

Back to Top