The Arctic circle boundary and the airy process

By K. Johansson. 2003

258 citations recorded by the source corpus.

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| �

Subfields: Atmospheric Science, Ecology

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Citations in the corpus

Determinantal Point Processes for Machine Learning (2012, background): “…community [Borodin and Olshanski, 2000, Borodin and Soshnikov, 2003, Borodin and Rains, 2005, Borodin et al., 2010, Burton and Pemantle, 1993, Johansson, 2002, 2004, 2005a, Okounkov, 2001, Okounkov and Reshetikhin, 2003, Shirai and Takahashi, 2000], and much progress has been made in…”
Height Fluctuations for the Stationary KPZ Equation (2014, result): “esses forthese models were determined. These statistical properties agreed between the models, but depended non-trivially on the type of initial data or geometry for the growth models, such as curved [6,10,21,22,60,61,68], ﬂat [10,21,23–25,43,48,73] or stationary [7–9,47,49,57,69]. All these results”
Eigenvalues of GUE Minors (2006, methodology): “It is shown in ( Joh05a ) that this process is a determinantal point process on N2 = {1,2,...,N }2,”
Stochastic six-vertex model (2014, background): “d vary 2R, then the (proper centered and scaled) random process H(Lx+L2=3;Ly;!) converges in distribution to the Airy 2 process (i.e. the top line of the Airy line ensemble). We refer to [PS2],[J2],[J3],[BF1] for similar statements for the interacting particle systems related to the determinantal po”
The Birth of a Random Matrix (2006, background): “Near a point inside the disordered region, one expects to see the unique translationinvariant ergodic Gibbs measure on stepped surfaces with given average slope (see e. g. [8], [11], [19] for theorems supporting these expectations).”
Random Skew Plane Partitions and the Pearcey Process (2005, background): “In a related but technically more involved context, the Airy process asymptotics was found by K. Johansson in [ 10 ].”
Large time asymptotics of growth models on space-like paths I: PushASEP (2007, unknown): “For the TASEP and PNG models, large time fluctuation results have already been obtained in the following cases: For the step initial condition the Airy2 process has been shown to occur in the scaling limit for fixed time [18; 14; 15], and more recently for tagged particle [13].”