Random Tiling Research Articles

COMBINATORIAL ANALYSIS OF THE DOMINOES SCHEME AND THE CASE OF FIXED MINIMAL FIGURE ON A DOMINO TILE

The dominoes scheme is defined as a scheme of random tiling with poliomino tiles with $r$ ends and $n$ figures from 0 to $(n-1)$ on the ends of tiles of all possible compositions with repetitions, regardless of their order. This scheme is studied, together with an analogous one, with a fixed minimal figure $r\ge m$ at the outcome of a random choice of a tile from a complete domino set, using the following ways of enumerative combinatorics: design of the enumeration procedure for the numbered outcomes of the scheme, determination of their number, solution of the corresponding numeration problem (i.e. establishment of one-to-one correspondence between numbers and types of the outcomes of the scheme), finding of their probabilistic distribution and modeling of their possible outcomes

One-Dimensionally Disordered Chiral Sorting by Racemic Tiling in a Surface-Confined Supramolecular Assembly of Achiral Tectons.

The aggregation of (pro)chiral/achiral molecules into crystalline structures at interfaces forms conglomerates, racemates, and solid solutions, comparable to known bulk phases. Scanning tunneling microscopy and Monte Carlo simulations were employed to uncover a distinct racemic phase, expressing 1D disordered chiral sorting through random tiling in surface-confined supramolecularly assembled achiral 4,4''-diethynyl-1,1':4',1''-terphenyl molecules. The configurational entropy of the 1D disordered racemic tiling phase was verified by analytical modeling, and found to lie between that of a perfectly ordered 2D racemate and a racemic solid solution.

One‐Dimensionally Disordered Chiral Sorting by Racemic Tiling in a Surface‐Confined Supramolecular Assembly of Achiral Tectons

AbstractThe aggregation of (pro)chiral/achiral molecules into crystalline structures at interfaces forms conglomerates, racemates, and solid solutions, comparable to known bulk phases. Scanning tunneling microscopy and Monte Carlo simulations were employed to uncover a distinct racemic phase, expressing 1D disordered chiral sorting through random tiling in surface‐confined supramolecularly assembled achiral 4,4′′‐diethynyl‐1,1′:4′,1′′‐terphenyl molecules. The configurational entropy of the 1D disordered racemic tiling phase was verified by analytical modeling, and found to lie between that of a perfectly ordered 2D racemate and a racemic solid solution.

The order-disorder evolution in quasicrystals through phason flips

Abstract The problem of incorporating phason flips in the structural investigation of aperiodic systems is still an open question and a challenge in crystallography. Phasons are understood both as atomic fluctuation and positional disorder of the quasicrystalline lattice. Popular correction to diffraction peaks' intensities takes the form of generalized Debye-Waller factor, assuming Gaussian distribution of fluctuations in the perpendicular space of higher-dimensional periodic lattice. Although proven to work in case of random tiling types of structures recent evidence indicates improper handling of peaks with high perpendicular space scattering vector whenever structure is far from random tiling regime. We introduce the concept of a series expansion of the characteristic function of the statistical distribution to properly correct the peaks' intensities with respect to phasonic fluctuations. Calculations are performed upon Penrose tiling. Such approximation of the structure factor works correctly even in cases for which the Debye-Waller correction fails. Even more we investigate transition to random tiling through phason flips by means of the statistical approach which results in interesting scaling properties (ordered → disordered → random → amorphous structure).

GUE corners limit of $q$-distributed lozenge tilings

We study asymptotics of $q$-distributed random lozenge tilings of sawtooth domains (equivalently, of random interlacing integer arrays with fixed top row). Under the distribution we consider each tiling is weighted proportionally to $q^{\mathsf{vol} }$, where $\mathsf{vol} $ is the volume under the corresponding 3D stepped surface. We prove the following Interlacing Central Limit Theorem: as $q\rightarrow 1$, the domain gets large, and the fixed top row approximates a given nonrandom profile, the vertical lozenges are distributed as the eigenvalues of a GUE random matrix and of its successive principal corners. Our results extend the GUE corners asymptotics for tilings of bounded polygonal domains previously known in the uniform (i.e., $q=1$) case. Even though $q$ goes to $1$, the presence of the $q$-weighting affects non-universal constants in our Central Limit Theorem.

Programmable disorder in random DNA tilings.

Scaling up the complexity and diversity of synthetic molecular structures will require strategies that exploit the inherent stochasticity of molecular systems in a controlled fashion. Here we demonstrate a framework for programming random DNA tilings and show how to control the properties of global patterns through simple, local rules. We constructed three general forms of planar network-random loops, mazes and trees-on the surface of self-assembled DNA origami arrays on the micrometre scale with nanometre resolution. Using simple molecular building blocks and robust experimental conditions, we demonstrate control of a wide range of properties of the random networks, including the branching rules, the growth directions, the proximity between adjacent networks and the size distribution. Much as combinatorial approaches for generating random one-dimensional chains of polymers have been used to revolutionize chemical synthesis and the selection of functional nucleic acids, our strategy extends these principles to random two-dimensional networks of molecules and creates new opportunities for fabricating more complex molecular devices that are organized by DNA nanostructures.

Bulk Universality for Random Lozenge Tilings Near Straight Boundaries and for Tensor Products

We prove that the asymptotic of the bulk local statistics in models of random lozenge tilings is universal in the vicinity of straight boundaries of the tiled domains. The result applies to uniformly random lozenge tilings of large polygonal domains on triangular lattice and to the probability measures describing the decomposition in Gelfand-Tsetlin bases of tensor products of representations of unitary groups. In a weaker form our theorem also applies to random domino tilings.

MCMC Methods for Drawing Random Samples From the Discrete-Grains Model of a Magnetic Medium

The discrete-grains model is a simple model for the distribution of “grains” on a magnetic medium. In this model, grains on the medium are taken to be one of four basic rectangular shapes (“tiles”)— $1 \times 1$ , $1 \times 2$ , $2 \times 1$ , and $2 \,\, \times \,\, 2$ . The magnetic medium is then modeled as an $N \times N$ square tiled by these four basic tiles. In this paper, we present Markov chain Monte Carlo (MCMC) methods for generating random tilings of an $N \times N$ square consisting only of the four basic tiles. To be precise, given a target probability distribution, e.g., the uniform distribution, on the space, $ {\mathcal S}_{N}$ , of all possible such tilings, we make use of the Metropolis–Hastings algorithm to design an MCMC method for sampling from a probability distribution arbitrarily close, in total variation distance, to the target distribution. We further extend this approach to enable sampling from probability distributions on certain subsets of $ {\mathcal S}_{N}$ , namely, those consisting of tilings, in which each kind of tile occurs a fixed number of times. We finally present some bounds and conjectures on the mixing times of the underlying Markov chains, which provide estimates of the amount of time taken by the MCMC methods to generate a random tiling sampled from the target probability distribution.

Gaussian asymptotics of discrete β $\beta $ -ensembles

We introduce and study stochastic $N$-particle ensembles which are discretizations for general-$\beta$ log-gases of random matrix theory. The examples include random tilings, families of non-intersecting paths, $(z,w)$-measures, etc. We prove that under technical assumptions on general analytic potential, the global fluctuations for such ensembles are asymptotically Gaussian as $N\to\infty$. The covariance is universal and coincides with its counterpart in random matrix theory. Our main tool is an appropriate discrete version of the Schwinger-Dyson (or loop) equations, which originates in the work of Nekrasov and his collaborators.

Predicting the outcome of the growth of binary solids far from equilibrium.

The growth of multicomponent structures in simulations and experiments often results in kinetically trapped, nonequilibrium objects. In such cases we have no general theoretical framework for predicting the outcome of the growth process. Here we use computer simulations to study the growth of two-component structures within a simple lattice model. We show that kinetic trapping happens for many choices of growth rate and intercomponent interaction energies, and that qualitatively distinct kinds of kinetic trapping are found in different regions of parameter space. In a region in which the low-energy structure is an "antiferromagnet" or "checkerboard," we show that the grown nonequilibrium structure displays a component-type stoichiometry that is different from the equilibrium one but is insensitive to growth rate and solution conditions. This robust nonequilibrium stoichiometry can be predicted via a mapping to the jammed random tiling of dimers studied by Flory, a finding that suggests a way of making defined nonequilibrium structures in experiment.

Honeycomb lattices with defects.

In this paper, we introduce a variant of the honeycomb lattice in which we create defects by randomly exchanging adjacent bonds, producing a random tiling with a distribution of polygon edges. We study the percolation properties on these lattices as a function of the number of exchanged bonds using an alternative computational method. We find the site and bond percolation thresholds are consistent with other three-coordinated lattices with the same standard deviation in the degree distribution of the dual; here we can produce a continuum of lattices with a range of standard deviations in the distribution. These lattices should be useful for modeling other properties of random systems as well as percolation.

Domino statistics of the two-periodic Aztec diamond

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 random surface. We consider the Aztec diamond with a two-periodic weighting which exhibits all three possible phases that occur in these types of models, often referred to as solid, liquid and gas. To analyze this model, we use entries of the inverse Kasteleyn matrix which give the probability of any configuration of dominoes. A formula for these entries, for this particular model, was derived by Chhita and Young (2014). In this paper, we find a major simplification of this formula expressing entries of the inverse Kasteleyn matrix by double contour integrals which makes it possible to investigate their asymptotics. In a part of the Aztec diamond, where the asymptotic analysis is simpler, we use this formula to show that the entries of the inverse Kasteleyn matrix converge to the known entries of the full-plane inverse Kasteleyn matrices for the different phases. We also study the detailed asymptotics of the inverse Kasteleyn matrix at both the ‘liquid–solid’ and ‘liquid–gas’ boundaries, and find the extended Airy kernel in the next order asymptotics. Finally we provide a potential candidate for a combinatorial description of the liquid–gas boundary.

Optimal transport between random measures

Nous analysons le probleme du transport optimal entre deux mesures aleatoires et equivariantes et demontrons des conditions qui garantissent l’existence d’une solution de type Monge. En outre, nous demontrons que l’equivariance apparait naturellement dans ce contexte en prouvant que les couplages optimaux classiques dans des ensembles bornes convergent vers le couplage optimal dans tout l’espace. Finalement nous demontrons des conditions suffisantes pour que le cout au sens $L^{p}$ soit fini en introduisant une metrique appropriee.

Edge fluctuations of limit shapes

In random tiling and dimer models we can get various limit shapes which gives the boundaries between different types of phases. The shape fluctuations at these boundaries give rise to universal limit laws, in particular the Airy process. We survey some models which can be analyzed in detail based on the fact that they are determinantal point processes with correlation kernels that can be computed. We also discuss which type of limit laws that can be obtained.

The Cusp-Airy process

At a typical cusp point of the disordered region in a random tiling model we expect to see a determinantal process called the Pearcey process in the appropriate scaling limit. However, in certain s ...

Asymptotics of symmetric polynomials with applications to statistical mechanics and representation theory

We develop a new method for studying the asymptotics of symmetric polynomials of representation-theoretic origin as the number of variables tends to infinity. Several applications of our method are presented: We prove a number of theorems concerning characters of infinite-dimensional unitary group and their $q$-deformations. We study the behavior of uniformly random lozenge tilings of large polygonal domains and find the GUE-eigenvalues distribution in the limit. We also investigate similar behavior for alternating sign matrices (equivalently, six-vertex model with domain wall boundary conditions). Finally, we compute the asymptotic expansion of certain observables in $O(n=1)$ dense loop model.

On the stability of a quasicrystal and its crystalline approximant in a system of hard disks with a soft corona.

Using computer simulations, we study the phase behavior of a model system of colloidal hard disks with a diameter σ and a soft corona of width 1.4σ. The particles interact with a hard core and a repulsive square-shoulder potential. We calculate the free energy of the random-tiling quasicrystal and its crystalline approximants using the Frenkel-Ladd method. We explicitly account for the configurational entropy associated with the number of distinct configurations of the random-tiling quasicrystal. We map out the phase diagram and find that the random tiling dodecagonal quasicrystal is stabilised by entropy at finite temperatures with respect to the crystalline approximants that we considered, and its stability region seems to extend to zero temperature as the energies of the defect-free quasicrystal and the crystalline approximants are equal within our statistical accuracy.

Lozenge Tilings and Hurwitz Numbers

We give a new proof of the fact that, near a turning point of the frozen boundary, the vertical tiles in a uniformly random lozenge tiling of a large sawtooth domain are distributed like the eigenvalues of a GUE random matrix. Our argument uses none of the standard tools of integrable probability. In their place, it uses a combinatorial interpretation of the Harish-Chandra/Itzykson-Zuber integral as a generating function for desymmetrized Hurwitz numbers.

Enumeration of octagonal tilings

Random tilings are interesting as idealizations of atomistic models of quasicrystals and for their connection to problems in combinatorics and algorithms. Of particular interest is the tiling entropy density, which measures the relation of the number of distinct tilings to the number of constituent tiles. Tilings by squares and 45° rhombi receive special attention as presumably the simplest model that has not yet been solved exactly in the thermodynamic limit. However, an exact enumeration formula can be evaluated for tilings in finite regions with fixed boundaries. We implement this algorithm in an efficient manner, enabling the investigation of larger regions of parameter space than previously were possible. Our new results appear to yield monotone increasing and decreasing lower and upper bounds on the fixed boundary entropy density that converge toward S∞=0.36021(3).

DEEP NEURAL NETWORKS FOR ABOVE-GROUND DETECTION IN VERY HIGH SPATIAL RESOLUTION DIGITAL ELEVATION MODELS

Abstract. Deep Learning techniques have lately received increased attention for achieving state-of-the-art results in many classification problems, including various vision tasks. In this work, we implement a Deep Learning technique for classifying above-ground objects within urban environments by using a Multilayer Perceptron model and VHSR DEM data. In this context, we propose a novel method called M-ramp which significantly improves the classifier’s estimations by neglecting artefacts, minimizing convergence time and improving overall accuracy. We support the importance of using the M-ramp model in DEM classification by conducting a set of experiments with both quantitative and qualitative results. Precisely, we initially train our algorithm with random DEM tiles and their respective point-labels, considering less than 0.1% over the test area, depicting the city center of Munich (25 km2). Furthermore with no additional training, we classify two much larger unseen extents of the greater Munich area (424 km2) and Dongying city, China (257 km2) and evaluate their respective results for proving knowledge-transferability. Through the use of M-ramp, we were able to accelerate the convergence by a magnitude of 8 and achieve a decrease in above-ground relative error by 24.8% and 5.5% over the different datasets.