[Paper Review] Random Domino Tilings and the Arctic Circle Theorem

William Jockusch, James Propp+1 more|arXiv (Cornell University)|Jan 13, 1998

Stochastic processes and statistical mechanicsMathematics163 citations

TL;DR

This paper proves the Arctic Circle Theorem, showing that in a random domino tiling of a large Aztec diamond, the central region (temperate zone) where tiles are randomly oriented is asymptotically circular with radius $ n/ackslash sqrt{2} $, while the outer four regions exhibit ordered, brickwork-like tilings. The proof uses a connection to the totally asymmetric exclusion process (TASEP) in discrete time, classifying its stationary measures to establish the circular boundary of the disordered region.

ABSTRACT

In this article we study domino tilings of a family of finite regions called Aztec diamonds. Every such tiling determines a partition of the Aztec diamond into five sub-regions; in the four outer sub-regions, every tile lines up with nearby tiles, while in the fifth, central sub-region, differently-oriented tiles co-exist side by side. We show that when n is sufficiently large, the shape of the central sub-region becomes arbitrarily close to a perfect circle of radius n/sqrt(2) for all but a negligible proportion of the tilings. Our proof uses techniques from the theory of interacting particle systems. In particular, we prove and make use of a classification of the stationary behaviors of a totally asymmetric one-dimensional exclusion process in discrete time.

Motivation & Objective

• To understand the large-scale structure of random domino tilings of Aztec diamonds.
• To explain why the disordered (temperate) region in such tilings is asymptotically circular.
• To establish that the boundary between the disordered central region and the ordered outer regions converges to a circle as the diamond size increases.
• To use stochastic processes, particularly the totally asymmetric exclusion process (TASEP), to model and analyze tiling dynamics.
• To demonstrate that the four outer regions correspond to distinct, phase-separated tiling patterns due to phase differences in the underlying lattice coloring.

Proposed method

• Model domino tilings using a height function that satisfies local Lipschitz constraints and boundary conditions.
• Use the shuffling algorithm to generate random tilings uniformly at random, based on particle dynamics on a lattice.
• Map the tiling process to a totally asymmetric exclusion process (TASEP) on $ \mathbb{Z} $, where particles move right with 50% probability if the site to the right is empty.
• Analyze the stationary behavior of the TASEP using translation-invariant probability measures and classify them based on particle density $ p $.
• Show that for $ p < 1/2 $, the only stationary measures are Bernoulli measures with fixed density, leading to elliptical boundaries in the tiling.
• Use the asymptotic behavior of the TASEP to derive the circular shape of the arctic circle, with radius $ n/\sqrt{2} $, by analyzing the limit of the exclusion process as $ p \to 0 $.

Experimental results

Research questions

• RQ1 What is the large-scale geometric structure of a random domino tiling of an Aztec diamond?
• RQ2 Why does the disordered central region in such tilings approach a perfect circle as the size increases?
• RQ3 How do the four outer regions of the tiling differ in their local tiling patterns, and why are they phase-separated?
• RQ4 What role does the totally asymmetric exclusion process (TASEP) play in modeling the dynamics of tiling generation?
• RQ5 Can the boundary between the ordered and disordered regions be rigorously shown to converge to a circle?

Key findings

• For any $ \epsilon > 0 $, all but an $ \epsilon $-fraction of random domino tilings of the Aztec diamond of order $ n $ have a temperate zone whose boundary lies within distance $ \epsilon n $ of the inscribed circle of radius $ n/\sqrt{2} $.
• The four outer regions of the tiling exhibit distinct, phase-separated brickwork patterns due to phase differences in the lattice coloring: top and bottom rows start with different colored squares, and left/right sides differ in vertical tile alignment.
• The only stationary translation-invariant measures for the TASEP with $ p < 1/2 $ are Bernoulli measures with fixed density $ p $, which correspond to the ordered tiling phases.
• The boundary of the disordered region is asymptotically circular, and the shape of the arctic circle is rigorously derived via the TASEP's stationary behavior.
• When $ p \to 0 $, the elliptical boundary of the TASEP's invariant measure converges to a parabolic arc, consistent with known results for the continuous-time exclusion process.
• The average height function on the Aztec diamond cannot be approximated by a linear function due to inconsistent boundary conditions, implying that local statistics cannot be homogeneous, which forces the existence of a disordered central region.