On the Shuffling Algorithm for Domino Tilings

Abstract

We study the dynamics of a certain discrete model of interacting particles that comes from the so called shuffling algorithm for sampling a random tiling of an Aztec diamond. It turns out that the transition probabilities have a particularly convenient determinantal form. An analogous formula in a continuous setting has recently been obtained by Jon Warren studying certain model of interlacing Brownian motions which can be used to construct Dyson's non-intersecting Brownian motion.We conjecture that Warren's model can be recovered as a scaling limit of our discrete model and prove some partial results in this direction. As an application to one of these results we use it to rederive the known result that random tilings of an Aztec diamond, suitably rescaled near a turning point, converge to the GUE minor process.

Multilevel Dyson Brownian motions via Jack polynomials

“…The special cases of such processes arise naturally in the study of two-dimensional statistical mechanics systems such as random stepped surfaces and various types of tilings (cf. [6,9,10,34]). …”

Anisotropic Growth of Random Surfaces in 2 + 1 Dimensions

“…The shuffling algorithm for domino tilings of Aztec diamonds introduced in [21] also fits into our formalism. The corresponding discrete time Markov chain is described in Section 2 below, and its equivalence to domino shuffling is established in the recent paper [39].…”

“…and with the densely packed initial condition x m k (n − m) = k − m − 1, the Markov chain P (n) ∆ discussed above is equivalent to the so-called shuffling algorithm on domino tilings of the Aztec diamonds that at time n produces a random domino tiling of the diamond of size n distributed according to the measure that assigns to a tiling the weight proportional to β raised to the number of vertical tiles, see [39].…”

Shuffling algorithm for boxed plane partitions

“…The continuous time Markov chain considered in [3] in detail can be viewed as the degeneration of P ± near a corner of the hexagon as a, b, c become large, and either a and b is substantially larger than the other two. It is worth noting that the shuffling algorithm for domino tilings of the Aztec diamonds also fits into the formalism of [3], see Section 2.6 of [3] and [26].…”