Random Tilings and Markov Chains for Interlacing Particles

#### A. Borodin, Patrik L. Ferrari

2018, v.24, Issue 3

ABSTRACT

We explain the relation between certain random tiling models and interacting
particle systems belonging to the anisotropic KPZ (Kardar\tire Parisi\tire Zhang)
universality class in $2+1$-dimensions. The link between these two \emph{a
priori} disjoint sets of models is a consequence of the presence of shuffling
algorithms that generate random tilings under consideration. To see the
precise connection, we represent both a random tiling and the corresponding
particle system through a set of non-intersecting lines, whose dynamics is
induced by the shuffling algorithm or the particle dynamics. The resulting class
of measures on line ensembles also fits into the framework of the Schur
processes.

Keywords: Markov chain, random tiling, shuffling algorithm, Schur process