# Is the Aztec Diamond chaotic?

Firstly, let me explain the context. An Aztec Diamond is a shape which looks like: (A rigorous definition can be found at Mathworld.)

The Aztec Diamond Theorem states that such a shape of order $n$ can be tiled in $2^{\frac{n(n+1)}{2}}$ ways using $2 \times 1$ dominoes. The Arctic Circle Theorem states that as $n$ becomes large, a random domino tiling of this Aztec diamond tends to have the following properties:

• The tiles outside a certain circle will be arranged homogeneously in a “brick-wall” pattern (these are the homogeneously coloured areas in the example below). They are denoted as frozen.
• For each corner, these patterns are distinct in orientation and phase.

This can be visualised as follows: Suppose the diamond’s squares are marked black and white in a chessboard pattern. Then assign a different colour (red, green, blue, yellow) to each domino tile, depending on whether the top, left, right, or bottom half rests on a black square. A homogeneous brick-wall tiling would then correspond to a homogeneous colours as all tiles within have the same orientation and phase. A typical tiling would then look like this: My question is about whether such a configuration can be described as chaotic. I regard chaos as both deterministic and having the property that when an initial parameter is slightly altered (in our case, the orientation/position of a domino), after a while the system will differ completely. So, if one alters a tile, for example on the border of the Arctic Circle for a large $n$, does that then impact the whole of the rest of the diagram? Once the corners are tiled, does that determine the rest of the diagram?

• I was just wondering about that. It would be good if it could be edited into the text at some point. Also what do the colors mean? If green represent 'standing' domino's and yellow 'lying' domino's, then what are blue and red? I thought there would only be two possibilities? – Vincent Jul 14 '17 at 9:04
• @Arthur duly noted and amended – Plato Jul 14 '17 at 9:07
• It might just be me not up to speed on the terminology here, but what does it mean that a tiling is frozen? – Arthur Jul 14 '17 at 9:08
• @Arthur That apart from a few possible tilings, all tilings have their corners tiled like shown in the image – Plato Jul 14 '17 at 9:11
• In case you hadn't found it yourself yet, here is the proof of the arctic circle theorem. It might contains some hints on your question: arxiv.org/pdf/math/9801068.pdf – Vincent Jul 14 '17 at 14:16

No, this lacks many properties of chaos. Chaos is a property of dynamical systems which change deterministically over time according to certain rules (maps or differential equations).

• The scenario is not a dynamical system. There is no notion of time. There is nothing that could be called an initial condition.
• There is no determinism. The scenario in question regards random samples from the huge set of all possible tilings (with $2^\frac{n(n+1)}{2}$ elements).

Once the corners are tiled, does that determine the rest of the diagram?

No. Each corner can only be tiled in two different ways (a tile lies in the corner or two tiles point into the corner). This makes $2^4$ possible ways to tile the corners of a given diamond. As $2^4 < 2^\frac{n(n+1)}{2}$, the arrangement of corners does not determine the entire arrangement.

[…] the chaos lies in that if we start at a corner and move towards the centre, we can predict that the tiles closer to the corner will be in that direction (fixed), but as we get closer to the centre and further from the corner, it becomes impossible to predict the direction of the tiles; like the weather.

Chaos is not only about uncertainty increasing. In case of a weather model, the uncertainty arises from the fact that small details of the initial state have a huge impact an evolved state. In your scenario, the uncertainty arises from the fact that you perform a walk on an area where certain regions are equipped with a higher variability than others. This in particular becomes apparent as the path of your walk is arbitrarily chosen. If you move from the centre to a corner or from one corner to another corner, the uncertainty behaves differently.

• "In case of a weather model, the uncertainty arises from the fact that small details of the initial state have a huge impact an evolved state" - can't one say this is true for the Aztec diamond. Placing the tile in the corner sideways yields a diagram which is very different than if we place two tiles pointing out? I realise that chaos is fundamentally connected to Dynamical Systems (In fact, that's where Lorenz first observed it), but I'm trying to see if one can notice a similar behaviour elsewhere. Do you think time can be considered as the number of tiles already placed? – Plato Jul 15 '17 at 11:20
• Also - I'm not trying to call you out or say that you are wrong, I just want to make sure I understand. (And cheers for the edit) – Plato Jul 15 '17 at 11:20
• @Plato: Do you think time can be considered as the number of tiles already placed? – You could consider it that way, but then you would still have a stochastic and not a deterministic process. Also, you only have one state that becomes more specific over time. Finally, you have a finite process. – Wrzlprmft Jul 15 '17 at 11:45
• What do you mean by "you only have one state that becomes more specific over time"? – Plato Jul 15 '17 at 11:49
• @Plato: Well, the situation you’re describing could be regarded as uncovering information about a given state, step by step. – Wrzlprmft Jul 15 '17 at 13:51

One should distinguish between disorder and chaos. Disorder involves a sufficiently fast decay of the pair correlation function between two spatial points as the distance between the points increases. Chaos applies to dynamical systems and assumes a positive Lyapunov exponent or, in words, an exponential divergence of typical trajectories with very close initial conditions. So we should call the "phase" inside the arctic circle disordered, rather than chaotic.