Random Tiling Research Articles

Combinatorics of the double-dimer model

We prove that the partition function for tripartite double-dimer configurations of a planar bipartite graph satisfies a recurrence related to the Desnanot-Jacobi identity from linear algebra. A similar identity for the dimer partition function was established nearly 20 years ago by Kuo and has applications to random tiling theory and the theory of cluster algebras. This work was motivated in part by the potential for applications in these areas. Additionally, we discuss an application to a problem in Donaldson-Thomas and Pandharipande-Thomas theory. The proof of our recurrence requires generalizing work of Kenyon and Wilson; specifically, lifting their assumption that the nodes of the graph are black and odd or white and even.

Random plane partitions and corner distributions

We explore some probabilistic applications arising in connections with $K$-theoretic symmetric functions. For instance, we determine certain corner distributions of random lozenge tilings and plane partitions. We also introduce some distributions that are naturally related to the corner growth model. Our main tools are dual symmetric Grothendieck polynomials and normalized Schur functions.

Decagonal random tilings and their quasi-unit cell cluster coverings

Electron microscopy images of decagonal quasicrystals obtained recently have been shown to be related to cluster coverings with a Hexagon–Bow–Tie decagon as single structural unit. Most decagonal phases show more complex structural orderings than models based on deterministic tilings like the Penrose tiling. We analyze different types of decagonal random tilings and their coverings by a Hexagon–Bow–Tie decagon.

Topological transformations in hyperuniform pentagonal two-dimensional materials induced by Stone-Wales defects

We discover two distinct topological pathways through which the pentagonal Cairo tiling (P5), a structural model for single-layer $AB_2$ pyrite materials, respectively transforms into a crystalline rhombus-hexagon (C46) tiling and random rhombus-pentagon-hexagon (R456) tilings, by continuously introducing the Stone-Wales (SW) topological defects. We find these topological transformations are controlled by the orientation correlations among neighboring $B$-$B$ bonds, and exhibit a phenomenological analogy of the (anti)ferromagnetic to paramagnetic transition in two-state Ising systems. Unlike the SW defects in hexagonal 2D materials such as graphene, which cause distortions, the defects in pentagonal 2D materials preserve the shape and symmetry of the fundamental cell of P5 tiling and are associated with a minimal energy cost, making the intermediate R456 tilings realizable metastable states at room temperature. Moreover, the intermediate structures along the two pathways are neither crystals nor quasicrystals, and yet these random tilings preserve hyperuniformity of the P5 or C46 crystal (i.e., the infinite-wavelength normalized density fluctuations are completely suppressed), and can be viewed as 2D analogs of disordered Barlow packings in three dimensions. The resulting 2D materials possess metal-like electronic properties, making them promising candidates for forming Schottky barriers with the semiconducting P5 material.

Extreme boundary conditions and random tilings

Standard statistical mechanical or condensed matter arguments tell us that bulk properties of a physical system do not depend too much on boundary conditions. Random tilings of large regions provide counterexamples to such intuition, as illustrated by the famous 'arctic circle theorem' for dimer coverings in two dimensions. In these notes, I discuss such examples in the context of critical phenomena, and their relation to 1+1d quantum particle models. All those turn out to share a common feature: they are inhomogeneous, in the sense that local densities now depend on position in the bulk. I explain how such problems may be understood using variational (or hydrodynamic) arguments, how to treat long range correlations, and how non trivial edge behavior can occur. While all this is done on the example of the dimer model, the results presented here have much greater generality. In that sense the dimer model serves as an opportunity to discuss broader methods and results. [These notes require only a basic knowledge of statistical mechanics.]

Domino Tilings of Cylinders: Connected Components under Flips and Normal Distribution of the Twist

We consider domino tilings of $3$-dimensional cubiculated regions. A three-dimensional domino is a $2\times 1\times 1$ rectangular cuboid. We are particularly interested in regions of the form $\mathcal{R}_N = \mathcal{D} \times [0,N]$ where $\mathcal{D} \subset \mathbb{R}^2$ is a fixed quadriculated disk. In dimension $3$, the twist associates to each tiling $\mathbf{t}$ an integer $\operatorname{Tw}(\mathbf{t})$. We prove that, when $N$ goes to infinity, the twist follows a normal distribution.
 A flip is a local move: two neighboring parallel dominoes are removed and placed back in a different position. The twist is invariant under flips. A quadriculated disk $\mathcal{D}$ is regular if, whenever two tilings $\mathbf{t}_0$ and $\mathbf{t}_1$ of $\mathcal{R}_N$ satisfy $\operatorname{Tw}(\mathbf{t}_0) = \operatorname{Tw}(\mathbf{t}_1)$, $\mathbf{t}_0$ and $\mathbf{t}_1$ can be joined by a sequence of flips provided some extra vertical space is allowed.
 Many large disks are regular, including rectangles $\mathcal{D} = [0,L] \times [0,M]$ with $LM$ even and $\min\{L,M\} \ge 3$. For regular disks, we describe the larger connected components under flips of the set of tilings of the region $\mathcal{R}_N = \mathcal{D} \times [0,N]$. As a corollary, let $p_N$ be the probability that two random tilings $\mathbf{T}_0$ and $\mathbf{T}_1$ of $\mathcal{D} \times [0,N]$ can be joined by a sequence of flips conditional to their twists being equal. Then $p_N$ tends to $1$ if and only if $\mathcal{D}$ is regular.
 Under a suitable equivalence relation, the set of tilings has a group structure, the {\em domino group} $G_{\mathcal{D}}$. These results illustrate the fact that the domino group dictates many properties of the space of tilings of the cylinder $\mathcal{R}_N = \mathcal{D} \times [0,N]$, particularly for large $N$.

Fluctuations of the arctic curve in the tilings of the Aztec diamond on restricted domains

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 that is the limit shape of the north polar region in the unrestricted model. We prove that the rescaled boundary of the north polar region in the restricted domain converges to the Airy$_2$ process conditioned to stay below a parabola with explicit continuous statistics and the finite dimensional distribution kernels. The limit is the hard-edge tacnode process which was first discovered in the framework of non-intersecting Brownian bridges. The proof relies on a random walk representation of the correlation kernel of the non-intersecting line ensemble which corresponds to a random tiling.

Highly symmetric aperiodic structures -INVITED

The symmetries of periodic structures are severely constrained by the crystallographic restriction. In particular, in two and three spatial dimensions, only rotational axes of order 1, 2, 3, 4 or 6 are possible. Aperiodic tilings can provide perfectly ordered structures with arbitrary symmetry properties. Random tilings can retain part of the aperiodic order as well the rotational symmetry. They offer a more flexible approach to obtain homogeneous structures with high rotational symmetry, and might be of particular interest for applications. Some key examples and their diffraction are discussed.

Dominos in hedgehog domains

We introduce a new class of discrete approximations of planar domains that we call “hedgehog domains”. In particular, this class of approximations contains two-step Aztec diamonds and similar shapes. We show that fluctuations of the height function of a random dimer tiling on hedgehog discretizations of a planar domain converge in the scaling limit to the Gaussian Free Field with Dirichlet boundary conditions. Interestingly enough, in this case the dimer model coupling function satisfies the same Riemann-type boundary conditions as fermionic observables in the Ising model. In addition, using the same factorization of the double-dimer model coupling function as in [18], we show that in the case of approximations by hedgehog domains the expectation of the double-dimer height function is harmonic in the scaling limit.

A method to obtain reproducible Ki-67 indices in lung adenocarcinoma.

Proliferative activity, evaluated from the Ki-67 index, is a strong prognostic factor in lung adenocarcinoma (LADC). Here, we optimised a procedure to measure the Ki-67 index and establish the best cut-off value. We examined 342 stage I LADCs for the immunohistochemical expression of Ki-67 using different antibodies, MIB1 and SP6. The results revealed the superior specificity of SP6; therefore, SP6 was used in subsequent analyses. Slides were scanned with a virtual slide system. Using software, tumour cells were counted in a whole tumour. Thereafter, the tumour was evenly subdivided into 0.25-mm2 tiles. The frequency of positive cells was counted in each tile of an invasive area or the whole tumour. We calculated the number of tumour cells required to produce a 95% confidence interval (CI) <0.05. Additionally, we calculated coverage probabilities (CP) using two different methods, counting any number or 200 cells per tile. The results showed that we could meet our goal by counting 2000 cells from 10 random tiles (200 cells each) in invasive areas. We successfully developed an optimal procedure for determination of the Ki-67 labelling index using an SP6 antibody, which provided CP>70% and CI of <0.05 in more than 90% of cases. Furthermore, we identified an optimal cut-off value of 0.12 with an alternative of 0.15, based on disease recurrence. This procedure and the cut-off values may be used in the routine pathological diagnosis of LADC.

Doubly periodic lozenge tilings of a hexagon and matrix valued orthogonal polynomials.

We analyze a random lozenge tiling model of a large regular hexagon, whose underlying weight structure is periodic of period 2 in both the horizontal and vertical directions. This is a determinantal point process whose correlation kernel is expressed in terms of non‐Hermitian matrix valued orthogonal polynomials (OPs). This model belongs to a class of models for which the existing techniques for studying asymptotics cannot be applied. The novel part of our method consists of establishing a connection between matrix valued and scalar valued OPs. This allows to simplify the double contour formula for the kernel obtained by Duits and Kuijlaars by reducing the size of a Riemann–Hilbert problem. The proof relies on the fact that the matrix valued weight possesses eigenvalues that live on an underlying Riemann surface M of genus 0. We consider this connection of independent interest; it is natural to expect that similar ideas can be used for other matrix valued OPs, as long as the corresponding Riemann surface M is of genus 0. The rest of the method consists of two parts, and mainly follows the lines of a previous work of Charlier, Duits, Kuijlaars and Lenells. First, we perform a Deift–Zhou steepest descent analysis to obtain asymptotics for the scalar valued OPs. The main difficulty is the study of an equilibrium problem in the complex plane. Second, the asymptotics for the OPs are substituted in the double contour integral and the latter is analyzed using the saddle point method. Our main results are the limiting densities of the lozenges in the disordered flower‐shaped region. However, we stress that the method allows in principle to rigorously compute other meaningful probabilistic quantities in the model.

Foreground detection by ensembles of random polygonal tilings

In this work a novel region-based approach for the detection of foreground in video sequences is presented. The model consists of an ensemble of layers or tilings, where each tiling represents, by means of randomly chosen parallelogram regions, the background of the scene. Currently, the image size of video surveillance cameras far exceeds one megapixel (more than 1024 × 768), and pixel-based proposals are poorly suited for near real-time ratios. Therefore, the analysis by pixel is replaced by an analysis by region, improving the final resolution by overlapping regions or parallelograms with different shapes and sizes. Thus, for each frame, each region estimates the probability of belonging to the foreground or background, to finally compute the consensus foreground mask among all the tilings. With this proposal, it is possible to detect the foreground in high resolution sequences, a process that is not feasible using pixel-level techniques. Several experiments have been carried out by employing a wide range of videos. A qualitative and quantitative comparison with the state-of-the-art algorithms is performed by using a well-known video dataset benchmark. The results show the feasibility of our proposal compared with higher resolution methods.

Stable quaternary Al59Cr23Fe8Si10 decagonal quasicrystal

We report a thermodynamically stable quaternary Al59Cr23Fe8Si10 decagonal quasicrystal (DQC), whose stability was verified by annealing at both high and low temperature for 168 or 360 h. The atomic scale structural features of the quasiperiodic plane were revealed based on high-angle annular dark-field scanning transmission electron microscopy images and the reasons for the stability of the Al59Cr23Fe8Si10 DQC are briefly discussed based on the structural characteristics. We argue that the 2 nm decagonal clusters with broken tenfold symmetry and their random tiling in the quasiperiodic plane of the Al59Cr23Fe8Si10 DQC enhance the stability.

Asymptotic Geometry of Discrete Interlaced Patterns: Part II

We study the asymptotic boundary of the liquid region of large random lozenge tiling models defined by uniformly random interlacing particle systems. In particular, we study a non-phase separating part of the boundary, i.e., a part of the boundary that does not border to a frozen phase. This is called the singular part of the boundary. We prove that isolated components of this boundary are lines and classify four different cases. Moreover, we show that the singular part of the boundary can have infinite one-dimensional Hausdorff measure. This has implications to the study of the free boundary problem arising in the variational problem studied by Kenyon and Okounkov and in a related work by De Silva and Savin.

Infinite-pressure phase diagram of binary mixtures of (non)additive hard disks.

One versatile route to the creation of two-dimensional crystal structures on the nanometer to micrometer scale is the self-assembly of colloidal particles at an interface. Here, we explore the crystal phases that can be expected from the self-assembly of mixtures of spherical particles of two different sizes, which we map to (additive or non-additive) hard-disk mixtures. We map out the infinite-pressure phase diagram for these mixtures using Floppy Box Monte Carlo simulations to systematically sample candidate crystal structures with up to 12 disks in the unit cell. As a function of the size ratio and the number ratio of the two species of particles, we find a rich variety of periodic crystal structures. Additionally, we identify random tiling regions to predict random tiling quasicrystal stability ranges. Increasing non-additivity both gives rise to additional crystal phases and broadens the stability regime for crystal structures involving a large number of large-small contacts, including random tilings. Our results provide useful guidelines for controlling the self-assembly of colloidal particles at interfaces.

Randomly Generated Identities

Benjamin et al. [1] introduced the idea of random tilings to prove a variety of identities relating to the Fibonacci numbers. Their method can be adapted to prove a range of familiar identities. We...

The application of graph theory for improving and rendering algorithm for finding the shortest path in the mathematical models of video games

Design of complex systems, study their properties and management require the development of a mathematical model. Study of the characteristics of the system using mathematical models is often the only way of studying complex systems and the solution of important practical problems.The application of graph theory to build mathematical models due to the possibility of describing them, a broad class of objects and processes. This gives you the ability to automate the search for optimal route, reach goals, network planning. One area of such applications is video games that require interface development, visualization of user actions and characters, development of the algorithm and the calculation of the moving actions of the characters in the game. Often used games in the genre Roguelike, the features of which is the creation of random levels, turn-based gameplay, tile (tile) or ASCII graphics. In this paper we propose a generalization of the algorithm for finding the shortest path algorithm A* with dynamic the ultimate goal of using the method of passage of all vertices in the weighted undirected graph and an implementation of this algorithm for visualization.

Phasonic Diffusion and Self-confinement of Decagonal Quasicrystals in Hyperspace

We introduce a novel simulation method that is designed to explore fluctuations of the phasonic degrees of freedom in decagonal colloidal quasicrystals. Specifically, we attain and characterise thermal equilibrium of the phason ensemble via Monte Carlo simulations with particle motions restricted to elementary phasonic flips. We find that, at any temperature, the random tiling ensemble is strongly preferred over the minimum phason-strain quasicrystal. Phasonic flips are the dominant carriers of diffusive mass transport in physical space. Sub- diffusive transients suggest cooperative flip behaviour on short time scales. In complementary space, particle mobility is geometrically restricted to a thin ring around the acceptance domain, resulting in self-confinement and persistent phasonic order.

A comparison of dynamic properties of a core-softened system of particles across glass transition, melting and random tiling formation

ABSTRACTSolid structures with different types of ordering (crystals, random tilings and glasses) are considered. We show that the short-range order structure of random tilings characterised by radial distribution functions looks very similar to the ones of glasses. However, the dynamical properties of random tilings look qualitatively identical to the ones of crystals. Because of this basing on radial distribution functions, only the system can be erroneously identified as glass or as random tiling. However, combining it with the dynamical properties, such as mean square displacement of stress autocorrelation function one can unambiguously distinguish glass and random tiling structure.

Tangent method for the arctic curve arising from freezing boundaries

In the paper Di Francesco and Guitter (2018 J. Phys. A: Math. Theor. 51 355201), the authors study the arctic curve arising in random tilings of some planar domains with an arbitrary distribution of defects on one edge. Using the tangent method they derive a parametric equation for portions of arctic curve in terms of an arbitrary piecewise differentiable function that describes the defect distribution. When this distribution presents ‘freezing’ intervals, other portions of arctic curve appear and typically have a cusp. These freezing boundaries can be of two types, respectively with maximal or minimal density of defects. Our purpose here is to extend the tangent method derivation of Di Francesco and Guitter (2018 J. Phys. A: Math. Theor. 51 355201) to include these portions, hence answering the open question stated in Di Francesco and Guitter (2018 J. Phys. A: Math. Theor. 51 355201).