Interlaced particle systems and tilings of the Aztec diamond

TL;DR

This paper introduces a weighted interlaced particle system modeling domino tilings of the Aztec diamond, providing exact marginal distributions and connecting to the GUE minor process, thus advancing understanding of tiling configurations.

Contribution

It defines a novel interlaced particle system for Aztec diamond tilings and derives exact distributions, linking tiling enumeration to random matrix theory.

Findings

Exact marginal distributions for the particle system are computed.

The number of tiling configurations is evaluated explicitly.

Connections to the GUE minor process are established.

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.