The Arctic Circle Theorem (Jockusch, Propp & Shor 1998): a uniformly random
domino tiling of the order-n Aztec diamond has a sharp phase boundary. Outside an inscribed
circle of radius n/√2, the tiling is deterministically "frozen" into one of four orientations.
Inside — the temperate zone — tiles are disordered. The boundary converges to an
exact circle as n→∞. This remarkable phase separation emerges purely from combinatorial randomness.