Random tilings and Markov chains for interlacing particles

TL;DR

This paper explores the connection between random tiling models and 2+1-dimensional anisotropic KPZ particle systems, revealing a unified framework through non-intersecting line ensembles and shuffling algorithms.

Contribution

It establishes a precise link between random tilings and interacting particle systems via non-intersecting line representations and Schur processes.

Findings

Unified representation of tilings and particle systems

Connection via non-intersecting line ensembles

Framework includes shuffling algorithms and Schur processes

Abstract

We explain the relation between certain random tiling models and interacting particle systems belonging to the anisotropic KPZ (Kardar-Parisi-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.