Asymptotic domino statistics in the Aztec diamond

Abstract

We study random domino tilings of the Aztec diamond with different weights for horizontal and vertical dominoes. A domino tiling of an Aztec diamond can also be described by a particle system which is a determinantal process. We give a relation between the correlation kernel for this process and the inverse Kasteleyn matrix of the Aztec diamond. This gives a formula for the inverse Kasteleyn matrix which generalizes a result of Helfgott. As an application, we investigate the asymptotics of the process formed by the southern dominoes close to the frozen boundary. We find that at the northern boundary, the southern domino process converges to a thinned Airy point process. At the southern boundary, the process of holes of the southern domino process converges to a multiple point process that we call the thickened Airy point process. We also study the convergence of the domino process in the unfrozen region to the limiting Gibbs measure.

General β‐Jacobi Corners Process and the Gaussian Free Field

“…First, the GFF is believed to be a universal scaling limit for various models of random surfaces in R 3 . Now the appearance of the GFF is rigorously proved for several models of random stepped surfaces; see Kenyon [54], Borodin-Ferrari [14], Petrov [71], Duits [28], Kuan [61], Chhita-Johansson-Young [22], and Borodin-Bufetov [11]. On the other hand, it is known that random matrix ensembles forˇD 2 can be obtained as a certain limit of stepped surfaces; see Okounkov-Reshetikhin [68], Johansson-Nordenstam [53], Fleming-Forrester-Nordenstam [34], Gorin [42,43], and Gorin-Panova [44].…”

The Generating Function for the Airy Point Process and a System of Coupled Painlevé II Equations

“…The thinned Airy point process arises not only as the large n limit of the thinned GUE, but also as limit point process in domino tilings of Aztec diamonds with different weights for horizontal and vertical dominoes [28].…”

Speed and fluctuations for some driven dimer models

“…The proof of [7,Theorem 2.9] involves showing the convergence of the entries of K −1 G L as L tends to infinity. Note that this limiting inverse Kasteleyn matrix is an inverse of a Kasteleyn matrix different from the one we considered in this paper given in (3.5); the two are gauge equivalent.…”