  The arctic circle theorem of Jockusch, Propp, and Shor asserts that uniformly
random domino tilings of an Aztec diamond of high order are frozen with
asymptotically high probability outside the "arctic circle" inscribed within
the diamond. A similar arctic circle phenomenon has been observed in the
limiting behavior of random square Young tableaux. In this paper, we show that
random domino tilings of the Aztec diamond are asymptotically related to random
square Young tableaux in a more refined sense that looks also at the behavior
inside the arctic circle. This is done by giving a new derivation of the
limiting shape of the height function of a random domino tiling of the Aztec
diamond that uses the large-deviation techniques developed for the square Young
tableaux problem in a previous paper by Pittel and the author. The solution of
the variational problem that arises for domino tilings is almost identical to
the solution for the case of square Young tableaux by Pittel and the author.
The analytic techniques used to solve the variational problem provide a
systematic, guess-free approach for solving problems of this type which have
appeared in a number of related combinatorial probability models.
