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?
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).
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).
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.
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.