Related papers: Crystallization of the Aztec diamond
Here we study the two-periodic weighted dimer model on the Aztec diamond graph. In the thermodynamic limit when the size of the graph goes to infinity while weights are fixed, the model develops a limit shape with frozen regions near…
We present a general approach for the study of dimer model limit shape problems via variational and integrable systems techniques. In particular we deduce the limit shape of the Aztec diamond and the hexagon for quasi-periodic weights…
The purpose of the present work is to provide a detailed asymptotic analysis of the $k\times\ell$ doubly periodic Aztec diamond dimer model of growing size for any $k$ and $\ell$ and under mild conditions on the edge weights. We explicitly…
We consider asymtotics of a domino tiling model on a class of domains which we call rectangular Aztec diamonds. We prove the Law of Large Numbers for the corresponding height functions and provide explicit formulas for the limit. For a…
In this paper we consider domino tilings of the Aztec diamond with doubly periodic weightings. In particular a family of models which, for any $ k \in \mathbb{N} $, includes models with $ k $ smooth regions is analyzed as the size of the…
We consider uniform random domino tilings of the restricted Aztec diamond which is obtained by cutting off an upper triangular part of the Aztec diamond by a horizontal line. The restriction line asymptotically touches the arctic circle…
Three phases of macroscopic domains have been seen for large but finite periodic dimer models; these are known as the frozen, rough and smooth phases. The transition region between the frozen and rough region has received a lot of attention…
This article has two main goals. First, it provides a self-contained exposition of the tangent plane method for the dimer model - a technique for analyzing arctic curves and limit shapes introduced by R. Kenyon and I. Prause (2020). Second,…
Recent advancements have been made to understand the statistics of the Aztec diamond dimer model under general periodic weights. In this work we define a model that breaks periodicity in one direction by combining two different two-periodic…
The thermodynamic limit is foundational to statistical mechanics, underlying our understanding of many-body phases. It assumes that, as the system size grows infinitely at fixed density of particles, unambiguous macroscopic phases emerge…
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…
We study the asymptotic behavior of random dimer coverings of the one-periodic Aztec diamond in random environment. We investigate quenched limit theorems for the height function and we extend annealed limit theorems that were recently…
In earlier work, Jockusch, Propp, and Shor proved a theorem describing the limiting shape of the boundary between the uniformly tiled corners of a random tiling of an Aztec diamond and the more unpredictable `temperate zone' in the interior…
We compute the algebraic equation for arctic curves of the Aztec diamond with a doubly (quasi-)periodic weight structure and obtain similar results for certain models of the hexagon. In particular, we determine the algebraic degree of such…
We consider the domino tilings of an Aztec diamond with a cut-off corner of macroscopic square shape and given size, and address the bulk properties of tilings as the size is varied. We observe that the free energy exhibits a third-order…
We study the octahedron relation (also known as the $A_{\infty}$ $T$-system), obeyed in particular by the partition function for dimer coverings of the Aztec Diamond graph. For a suitable class of doubly periodic initial conditions, we find…
We consider the dimer model on the Aztec diamond with Fock's weights, which is gauge equivalent to the model with any choice of positive weight function. We prove an explicit, compact formula for the inverse Kasteleyn matrix, thus extending…
We show there is a last path at the rough smooth boundary of the two-periodic Aztec diamond with parameter $a\in (0,1)$ that, suitably rescaled, converges to the Airy process, under the condition that $a$ tends to zero as the size of the…
In Bain [J. Math. Phys. 64, 023301 (2023)], we found asymptotics of one-point correlation functions of the two-periodic weighted Aztec diamond in the mesoscopic limit, where the linear size of the ordered region is of the same order as the…
Domino tilings of Aztec diamonds are known to exhibit an arctic phenomenon, namely a separation between frozen regions (in which all the dominoes have the same orientation) and a central disordered region (where dominoes are found without…