Arctic circles, domino tilings and square Young tableaux

TL;DR

This paper reveals a deep connection between random domino tilings of Aztec diamonds and square Young tableaux, providing a new derivation of their limiting shapes and demonstrating a unified approach to related combinatorial models.

Contribution

It introduces a refined analysis linking domino tilings and Young tableaux, using large-deviation techniques to solve associated variational problems systematically.

Findings

Asymptotic relation between domino tilings and Young tableaux inside the arctic circle.

New derivation of the limiting shape of domino tilings using large-deviation methods.

Unified approach to solving variational problems in combinatorial probability models.

Abstract

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.