Interlaced Particle Systems and Tilings of the Aztec Diamond

Abstract

Motivated by the problem of domino tilings of the Aztec diamond, a weighted particle system is defined on N lines, with line j containing j particles. The particles are restricted to lattice points from 0 to N , and particles on successive lines are subject to an interlacing constraint. It is shown that marginal distributions for this particle system can be computed exactly. This in turn is used to give unified derivations of a number of fundamental properties of the tiling problem, for example the evaluation of the number of distinct configurations and the relation to the GUE minor process. An interlaced particle system associated with the domino tiling of a certain half Aztec diamond is similarly defined and analyzed.

Domino Shuffling on Novak Half-Hexagons and Aztec Half-Diamonds

“…Order 100 half-hexagon, as an interlacing particle process whose domino shuffling algorithm is identical, in a certain sense, to that on the half-hexagon. Our Aztec half-diamonds are similar but not identical to the half Aztec diamond of [FF11].…”

Asymptotics of random domino tilings of rectangular Aztec diamonds

“…Thus, the frozen boundary is given by the equation p2χ ´1q 2 `p2κ ´1q 2 ´1 " 0, as expected. This domain was considered in [22] and subsequently in [37].…”

Asymptotic domino statistics in the Aztec diamond

“…Indeed, [21,22] showed that the law of the particles associated to the tiling is given by the Airy process and that the position of the last particle is given by the Tracy Widom distribution, F 2 ; see, for example, [1]. Furthermore, Johansson and Nordenstam [24] showed that the distribution of these particles becomes the GUE minor process at the intersection of the liquid region and the boundary of the Aztec diamond while Fleming and Forrester [17] obtained similar results for a certain half Aztec diamond.…”