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"The arctic circle" By Vincent Beffara

An other occurrence of the coexistence of two very different phases in a simple system is the following system, modeling the arrangement of molecules in a container. Consider a finite region formed of squares of the planar grid, and try to cover it by disjoint /dominos/ (a domino is just a 2x1 rectangle). Sometimes it is impossible - think of a domain composed of an odd number of squares; sometimes there is a unique way to do it - think of a long and thin rectangle. But if the chosen domain is “nice”, then typically there is a very large number of ways to do it. What does a typical covering by dominos look like? Do the dominos tend to align, or are they completely disordered? As it turns out, for certain domains, the two behaviors occur at different locations, and the boundary between them, separating the ordered and disordered phases, is known as the “arctic circle” (because it separates a frozen region from a liquid one). Understanding why there is such a brutal phase transition and where it occurs, although the model itself looks extremely simple, involves mathematics from very different fields, from probability and combinatorics to algebraic geometry; but quite a bit can be understood using only basic tools.

Domino tiling

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