The arctic circle boundary and the Airy process

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. 1. Introduction and results. Domino tilings of the Aztec diamond were introduced in [8, 9]. Asymptotic properties of random domino tilings of the Aztec diamond have been studied in [5, 12, 15]. In particular, in [12] the existence of the so-called arctic circle was proved. The arctic circle is the asymptotic boundary of the disordered so-called temperate region of the tiling. Outside this boundary the tiling forms a completely regular brick wall pattern. The methods in [12] combined with the results in [13] show that the fluctuations of the point of intersection of the boundary of the temperate region with a line converge to the Tracy–Widom distribution of random matrix theory. In this paper we extend this result to show that the fluctuations of the boundary around the arctic circle converges to the Airy process introduced in [23]. The paper is a continuation of the approach used in [14] and [15], where certain point processes with determinantal correlation functions [24] and the Krawtchouk ensemble, were used. We will use the general techniques developed in [16] and investigate an extended point process which also has determinantal correlation functions given by a kernel, which we call the extended Krawtchouk kernel. The Aztec diamond, An, of order n is the union of all lattice squares [m, m+1]×[l, l+1], m, l 2 Z, that lie inside the region {(x1, y1);|x1|+|y1| �