This series of pictures depicts the "Arctic circle" phenomenon. Let us try to tile this black diamon shape, with the dark grey, and light grey tiles shown in the top left and right corners. 

The dark grey shape consists of two squares glued together horizontally, while the light grey shape consists of two squares glued together vertically. 

For each size of the diamond, there are a certain number of tilings. Let us pick one uniformly at random. When we scale up the diamond, we see the "arctic circle" appear.

Although the tilings are drawn randomly, the corners of the tiling are not random at all, but rather tiled with a single species of tiles. The tiling is "frozen" outside of the "arctic circle", and random-looking, "liquid" on the inside. This phenomenon occurs because the shape of the diamond has the consequence that there are very few tilings with a dark grey piece near the left corner. Almost all the tilings have light grey pieces there, and when one is chosen at random, it has light grey pieces there with very high probability. 

A similar phenomenon appears with square Young tableaux. A square young tableau of size n is a n x n square in which each cell is assigned an integer from 1 to n^2, in such a way that all the rows and all the columns are increasing. Like with the tilings above, we can draw square Young tableau at random, increasing their size. We picture here the number in each cell by a share of grey, where white corresponds to the smallest number and black to the largest.

Or picturing 10 level curves

As the size of the tableau grows, the typical random configuration does not look "random" at all. 

A closely related example is given by jump sequences. Imagine n particles constrained to stay on 2n possible sites. No two particles can be simultaneously on the same site. The particles all start at the positions 1, 2, ..., n and end at the positions n + 1, ..., 2n, and can only move to the right (to a position with higher index). In between they move randomly within the constraints above. As no two particles can sit at the same time at the same site, their trajectories never cross, and they will reach their final position after exactly n^2 jumps. The trajectories of the system are in fact in bijection with the square Young tableaux discussed above, see the paper linked below for details. We can again plot a random trajectory of the system, for increasingly large n.

We see a perfect circle appear, outside of which particles are forzen on their site.

The paper Arcic circles, domino tilings and square Young tableaux by D. Romik was the source of inspiration for this post. The code to generate the Aztec diamond tilings was adapted from this github repo by M. Jasperse.

Dall-e's interpretation of the arctic phenomenon in dimer tilings of the Aztec diamond, jump sequences, etc...