The arctic circle boundary and the Airy process

TL;DR

This paper demonstrates that the rescaled boundary of the north polar region in the Aztec diamond converges to the Airy process, using determinantal point processes and proving a conjecture about the tiling structure.

Contribution

It establishes the convergence of the Aztec diamond boundary to the Airy process and proves a conjecture on the tiling structure at the center.

Findings

Boundary converges to the Airy process

Proves a version of Propp's conjecture

Uses extended Krawtchouk kernel in proof

Abstract

We prove that the, appropriately rescaled, boundary of the north polar region in the Aztec diamond converges to the Airy process. The proof uses certain determinantal point processes given by the extended Krawtchouk kernel. We also prove a version of Propp's conjecture concerning the structure of the tiling at the center of the Aztec diamond.