Random tilings and Markov chains for interlacing particles

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.

34 pages, 20 figures, LaTeX. To be printed with colors

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Cited by in corpus (10)

• TASEP and generalizations: Method for exact solution
• Perfect sampling algorithm for Schur processes
• The hard-edge tacnode process for Brownian motion
• Schur function at generic points and limit shape of perfect matchings on contracting square hexagon lattices with piecewise boundary conditions
• Fluctuations of the Arctic curve in the tilings of the Aztec diamond on restricted domains
• Pearcey universality at cusps of polygonal lozenge tiling
• The domino shuffling height process and its hydrodynamic limit
• Asymptotics of pure dimer coverings on rail-yard graphs
• Discrete dynamics in cluster integrable systems from geometric $R$-matrix transformations
• Limit shape of perfect matchings on contracting bipartite graphs