In combinatorial

Combinatorics

Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...

 mathematics, an Aztec diamond of order n consists of all squares of a square lattice whose centers (x,y) satisfy |x| + |y| ≤ n. Here n is a fixed integer, and the square lattice consists of unit squares with the origin as a vertex of 4 of them, so that both x and y are half an odd integer.

The Aztec diamond theorem states that the number of domino tiling

Domino tiling

A domino tiling of a region in the Euclidean plane is a tessellation of the region by dominos, shapes formed by the union of two unit squares meeting edge-to-edge...

s of the Aztec diamond of order n is 2^n(n+1)/2. The arctic circle theorem

Arctic circle theorem

In mathematics, the arctic circle theorem of states that random domino tilings of a large Aztec diamond tend to be frozen outside a certain "arctic circle"....

says that a random tiling of a large Aztec diamond tends to be frozen outside a certain circle.