Related papers: From Aztec diamonds to pyramids: steep tilings
On a finite weighted graph, the dimer model is a probability measure on its dimer covers, that assigns to any cover a probability proportional to the product of the weights of its edges. For planar bipartite graphs, dimer correlations are…
In this thesis, we consider domino tilings of three-dimensional regions, especially those of the form $\mathcal{D} \times [0,N]$. In particular, we investigate the connected components of the space of tilings of such regions by flips, the…
We study classical dimers on two-dimensional quasiperiodic Ammann-Beenker (AB) tilings. Despite the lack of periodicity we prove that each infinite tiling admits 'perfect matchings' in which every vertex is touched by one dimer. We…
This is a review of recent results on the integrable structure of the ordinary and modified melting crystal models. When deformed by special external potentials, the partition function of the ordinary melting crystal model is known to…
Exploring nonminimal-rank quasicrystals, which have symmetries that can be found in both periodic and aperiodic crystals, often provides new insight into the physical nature of aperiodic long-range order in models that are easier to treat.…
A domino covering of a board is saturated if no domino is redundant. We introduce the concept of a fragment tiling and show that a minimal fragment tiling always corresponds to a maximal saturated domino covering. The size of a minimal…
The number of domino tilings of a region with reflective symmetry across a line is combinatorially shown to depend on the number of domino tilings of particular subregions, modulo 4. This expands upon previous congruency results for domino…
We build a new perspective to count perfect matchings of a given graph. This idea is motivated by a construction on the relative cohomology group of surfaces. As an application of our theory, we reprove the celebrated Aztec Diamond theorem,…
In this paper, we introduce and study two new classes of commutative rings, namely semi transitional rings and transitional rings, which extend several classical ideas arising from rings of continuous functions and their variants. A general…
We investigate certain measures induced by families of non-intersecting paths in domino tilings of the Aztec diamond, rhombus tilings of an abc-hexagon, a dimer model on a cylindrical brick lattice and a growth model. The measures obtained,…
As a continuation to our previous work [9, 10], we consider the domino tiling problem with impurities. (1) if we have more than two impurities on the boundary, we can compute the number of corresponding perfect matchings by using the…
The present paper describes the $W$--geometry of the Abelian finite non-periodic (conformal) Toda systems associated with the $B,C$ and $D$ series of the simple Lie algebras endowed with the canonical gradation. The principal tool here is a…
Let $A$ be an expanding matrix on ${\Bbb R}^s$ with integral entries. A fundamental question in the fractal tiling theory is to understand the structure of the digit set ${\mathcal D}\subset{\Bbb Z}^s$ so that the integral self-affine set…
A new family of decagonal quasiperiodic tilings are constructed by the use of generalized point substitution processes, which is a new substitution formalism developed by the author [N. Fujita, Acta Cryst. A 65, 342 (2009)]. These tilings…
The edge-to-edge tilings of the sphere by congruent quadrilaterals of Type $a^2bc$ are classified as $3$ classes: a sequence of two-parameter families of $2$-layer earth map tilings with $2n$ $(n\ge3)$ tiles, a one-parameter family of…
An $N$-tiling of triangle $ABC$ by triangle $T$ (the `tile') is a way of writing $ABC$ as a union of $N$ copies of $T$ overlapping only at their boundaries. Let the tile $T$ have angles $(\alpha,\beta,\gamma)$, and sides $(a,b,c)$. This…
Matrix valued orthogonal polynomials (MVOP) appear in the study of doubly periodic tiling models. Of particular interest is their limiting behavior as the degree tends to infinity. In recent years, MVOP associated with doubly periodic…
Let $n,d\in \mathbb{N}$ and $n>d$. An $(n-d)$-domino is a box $I_1\times \cdots \times I_n$ such that $I_j\in \{[0,1],[1,2]\}$ for all $j\in N\subset [n]$ with $|N|=d$ and $I_i=[0,2]$ for every $i\in [n]\setminus N$. If $A$ and $B$ are two…
A finite collection $P$ of finite sets tiles the integers iff the integers can be expressed as a disjoint union of translates of members of $P$. We associate with such a tiling a doubly infinite sequence with entries from $P$. The set of…
To understand an aperiodic tiling (or a quasicrystal modeled on an aperiodic tiling), we construct a space of similar tilings, on which the group of translations acts naturally. This space is then an (abstract) dynamical system. Dynamical…