Related papers: Random Domino Tilings and the Arctic Circle Theore…
In this paper, we continue the study of domino-tilings of Aztec diamonds. In particular, we look at certain ways of placing ``barriers'' in the Aztec diamond, with the constraint that no domino may cross a barrier. Remarkably, the number of…
We study the asymptotic behavior of random domino tilings of the Aztec diamond of size $M$ in a random environment, where the environment is a one-periodic sequence of i.i.d. random weights attached to domino positions (i.e., to the edges…
We consider a generating function of the domino tilings of an Aztec rectangle with several boundary unit squares removed. Our generating function involves two statistics: the rank of the tiling and half number of vertical dominoes as in the…
We obtain precise asymptotics for the weighted number of domino tilings of an L-shaped subset of the Aztec diamond, obtained by removing an approximate rectangle in a corner of the Aztec diamond. By tuning the size of the removed corner, we…
We consider the six-vertex model at its free-fermion point with domain wall boundary conditions, which is equivalent to random domino tilings of the Aztec diamond. We compute the scaling limit of a particular non-local correlation function,…
We study tilings of regions in the square lattice with L-shaped trominoes. Deciding the existence of a tiling with L-trominoes for an arbitrary region in general is NP-complete, nonetheless, we identify restrictions to the problem where it…
We consider domino tilings of $3$-dimensional cubiculated regions. A three-dimensional domino is a 2x2x1 rectangular cuboid. We are particularly interested in regions of the form $R_N = D \times [0,N]$ where $D$ is a fixed quadriculated…
The inverse Kasteleyn matrix of a bipartite graph holds much information about the perfect matchings of the system such as local statistics which can be used to compute local and global asymptotics. In this paper, we consider three…
We study the enumeration of off-diagonally symmetric domino tilings of odd-order Aztec diamonds in two directions: (1) with one boundary defect, and (2) with maximally-many zeroes on the diagonal. In the first direction, we prove a symmetry…
Random domino tilings of the Aztec diamond shape exhibit interesting features and some of the statistical properties seen in random matrix theory. As a statistical mechanical model it can be thought of as a dimer model or as a certain…
For various sets of tiles, we count the ways to tile an Aztec diamond of order $n$ using tiles from that set. The resulting function $f(n)$ often has interesting behavior when one looks at $n$ and $f(n)$ modulo powers of 2.
The problem of counting tilings of a plane region using specified tiles can often be recast as the problem of counting (perfect) matchings of some subgraph of an Aztec diamond graph A_n, or more generally calculating the sum of the weights…
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…
We analyze domino tilings of the two-periodic Aztec diamond by means of matrix valued orthogonal polynomials that we obtain from a reformulation of the Aztec diamond as a non-intersecting path model with periodic transition matrices. In a…
Based on a bijection between domino tilings of an Aztec diamond and non-intersecting lattice paths, a simple proof of the Aztec diamond theorem is given in terms of Hankel determinants of the large and small Schr\"oder numbers.
Di Francesco conjectured in 2021 that the number of domino tilings of a certain family of regions -- called Aztec triangles -- on the square lattice is given by a product formula reminiscent of the one giving the number of alternating sign…
Random tilings of the two-periodic Aztec diamond contain three macroscopic regions: frozen, where the tilings are deterministic; rough, where the correlations between dominoes decay polynomially; smooth, where the correlations between…
We explore the connections between the well-studied Aztec Diamond graphs and a new family of graphs called the Half-Hexagons, discovered by Jonathan Novak. In particular, both families of graphs have very simple domino shuffling algorithms,…
We study the large-scale geometry of t-surfaces -- pairs of perfect t-embeddings and their associated origami maps -- arising from dimer models on Aztec diamonds with periodic edge weights. We prove that these t-surfaces converge to…
In this paper we consider domino tilings of bounded regions in dimension $n \geq 4$. We define the twist of such a tiling, an elements of ${\mathbb{Z}}/(2)$, and prove it is invariant under flips, a simple local move in the space of…