Statistical mechanics, structure of matter Exactly solved models Theorem

Arctic circle theorem

Statement

Let $n \ge 1$ be an integer. The Aztec diamond $A_n$ of order $n$ is the union of the unit squares $[a, a+1] \times [b, b+1]$ with $a, b \in \Z$ inside the region $\abs{x} + \abs{y} \le n + 1$ . Fix a domino tiling of $A_n$ . Its four polar regions are the regions next to the four corners of $A_n$ in which all dominoes are parallel and form a brick-wall pattern. The rest of $A_n$ is the temperate region. Let $T_n$ be a domino tiling of $A_n$ chosen uniformly at random. Let $E_{n, \varepsilon}$ be the event that the boundary of the temperate region of $T_n$ , scaled by $1/n$ , lies within distance $\varepsilon$ of the circle $x^2 + y^2 = 1/2$ . Then for every $\varepsilon > 0$ , the probability of $E_{n, \varepsilon}$ tends to $1$ as $n \to \infty$ .

Remarks

The circle $x^2 + y^2 = 1/2$ is the circle inscribed in the square $\abs{x} + \abs{y} \le 1$ , the scaled limit of $A_n$ . So with high probability the tiling is frozen outside the inscribed circle and disordered inside it. The theorem is due to Jockusch, Propp and Shor. Their proof uses domino shuffling, a random growth procedure that builds a uniform tiling of $A_{n+1}$ from one of $A_n$ , and analyses the polar regions as an interacting particle system. The number of domino tilings of $A_n$ is $2^{n(n+1)/2}$ , a result of Elkies, Kuperberg, Larsen and Propp. Later work found the fluctuations of the boundary, which are governed by the Airy process.

Going deeper

For a student

It is a model case for the variational principle of Cohn, Kenyon and Propp for dimer models. It shows that boundary conditions can affect a dimer model deep in the bulk, unlike many short-range models. Similar arctic curves appear for random lozenge tilings of hexagons.

Reading path

To understand this page from the ground up, read these 2 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Domino tiling concept
Arctic circle theorem statement

Tag	`ST1PFA7B8A`
URI	`m2://ST1PFA7B8A`
Kind	statement, theorem
Writing	W0 stub
MSC	82B20, 05B45, 60C05
Status	This copy of the site has no build data. Run `m2 status ST1PFA7B8A`.
About	Domino tiling
Uses	Domino tiling
Changed	3 November 2026