Random Tiling Research Articles

Confirmation of the random tiling hypothesis by polar calculus

Confirmation of the random tiling hypothesis by polar calculus

Nearly Ideal Random Tiling with Dodecagonal Symmetry from Pentablock Quarterpolymers of the AB1CB2D Type.

Two-dimensional tiling manners as cross-sectional views of cylindrical domain assembly formed by pentablock quarterpolymers of the AB1CB2D type in bulk were investigated. Several binary and ternary blends from three mother polymers having different ϕB1/ϕB2 ratios (ϕB1 and ϕB2 are the volume fractions of the B1 and B2 blocks, respectively) represent nonperiodic but ordered triangle/square tilings, where the N3/N4 ratios (N3 and N4 are the numbers of triangles and squares in the observed area, respectively) are all close enough to the theoretical value of 4/√3 ≑ 2.31 for the dodecagonal quasicrystalline (DDQC) state, irrespective of the total number of polygons. The TEM images, having almost the same N3/N4 ratios, were proved to show 4- and 6-fold symmetries in terms of the angular appearance of equilateral polygon sides via image analyses. Among them, a ternary blend showed a nearly ideal random tiling pattern that is almost equivalent to the theoretically predicted tiling by SCFT. Moreover, the magnitude of phason strain estimated for a TEM image from the ternary blend was proved to be quite small when the observing area is narrow, while it deviates from the ideal quasicrystalline tiling with an increasing number of vertices in the observing area.

Pearcey universality at cusps of polygonal lozenge tilings

AbstractWe study uniformly random lozenge tilings of general simply connected polygons. Under a technical assumption that is presumably generic with respect to polygon shapes, we show that the local statistics around a cusp point of the arctic curve converge to the Pearcey process. This verifies the widely predicted universality of edge statistics in the cusp case. Together with the smooth and tangent cases proved by Aggarwal‐Huang and Aggarwal‐Gorin, these are believed to be the three types of edge statistics that can arise in a generic polygon. Our proof is via a local coupling of the random tiling with nonintersecting Bernoulli random walks (NBRW). To leverage this coupling, we establish an optimal concentration estimate for the tiling height function around the cusp. As another step and also a result of potential independent interest, we show that the local statistics of NBRW around a cusp converge to the Pearcey process when the initial configuration consists of two parts with proper density growth, via careful asymptotic analysis of the determinantal formulas.

Artificial Intelligence in the Detection of Fatty Liver Disease

Background: The current gold standard for quantification of the fat content of the liver (i.e., nonalcoholic fatty liver disease, NAFLD) is the visual microscopic inspection of liver biopsies by pathologists. Percentage of macrosteatosis (%MaS), used in determining NAFLD diagnosis, is vital in determining the transplant suitability of a donor liver. A major limitation of this method is inevitable human error which causes interobserver variability and overestimation of %MaS which could cause a potential discard of donor livers. We hypothesize that artificial intelligence (AI) can assist pathologists in providing a more objective and accurate measure of %MaS.
 Methods: Our literature review identified HALO (image analysis) and U-Net (deep-learning) as AI programs currently available for high-accuracy %MaS calculation in liver biopsies. We compared the pathologist-reported %MaS from de-novo liver transplant (LT) biopsy samples taken 2h post-reperfusion to the %MaS calculated by HALO and/or U-Net (Fig. 1). 250 patients had undergone de novo LT at Indiana University between 2019-2020, and 211 of these patients had sufficient data (liver enzyme tests, biopsy results) to be included. Each biopsy sample was digitized into 5 random non-overlapping tiles at 20x magnification using a Leica microscope. Early allograft dysfunction (EAD) is defined by the presence of at least one of the following: (i) INR >1.6 on postoperative day (POD) 7, (ii) total bilirubin >10 mg/dL on POD7, or (iii) AST/ALT >2000 IU/L within the first 7 days following LT.
 Results: Our literature review identified that both HALO and U-Net estimated the %MaS in liver allograft biopsies significantly lower than pathologists' estimation (Fig.1). Of 211 included patients, 46 (21.8%) had EAD. In this ongoing project, we found U-Net to have a 97.3% training accuracy with eight epochs (2000 biopsy images each). Tiles from the first ten patients are being analyzed by HALO and/or U-Net to calculate an average %MaS for each patient. These calculations will be compared to the %MaS estimation made by pathologists.
 Conclusions: The rapidly evolving field of AI is emerging as a promising method in the quantification of the fat content of the liver with increased accuracy. AI will therefore help pathologists and transplant surgeons to determine liver transplant viability and better predict EAD in transplant patients.

Tuning the stability of a model quasicrystal and its approximants with a periodic substrate.

Quasicrystals and their periodic approximants are complex crystalline phases. They have now been observed in many metallic alloys, soft matter systems, and particle simulations. In recent experiments of thin-film perovskites on solid substrates, the type of complex phase was found to change depending on thermodynamic conditions and the type of substrate used. Here, we investigate the effect of a substrate on the relative thermodynamic stability of a two-dimensional model quasicrystal and its approximants. Our simulation model is particles interacting via the Lennard-Jones-Gauss potential. Our numerical methods are molecular dynamics simulations and free energy calculations that take into account phason flips explicitly. For substrates interacting weakly with the particles, we observe an incommensurate-commensurate transition, in which a continuous series of quasicrystal approximants locks into a small number of approximants. Interestingly, we observe that the 3/2 approximant exhibits phason mode fluctuations in thermodynamic equilibrium. Such fluctuations are reminiscent of random tiling and a phenomenon usually associated only with quasiperiodic order. For stronger substrates, we find an enhancement of the stability of the dodecagonal quasicrystal and variants of square lattices. We explain all observed phenomena by the interplay of the model system with the substrate. Our results demonstrate that designing novel complex periodic and quasiperiodic structures by choice of suitable substrates is a promising strategy.

Tactile perception of randomness: Effect of varying stimulus size and participants age.

We investigated participants' ability to differentiate between random and organized two-dimensional tactile tiles with embossed dots and examined how this ability varies with size and participant age. Four experiments were conducted to evaluate the effect of these variations on participants' capacity to utilize touch in identifying which of two stimuli exhibited greater randomness. Participants were instructed to explore embossed tiles using both hands. The tiles had varying levels of randomness from organized to random sets. In Experiments 1, 2, and 4, the sets were of equal size, while in Experiment 3, they differed in size. Results revealed a significant difference between the random and organized sets, with random stimuli being more easily discernible. These findings suggest that touch can be utilized to discern random patterns on tactile maps or displays. However, older participants encountered difficulties making this distinction, indicating similarities between vision and touch in perceiving randomness.

Spectral cocycle for substitution tilings

AbstractThe construction of a spectral cocycle from the case of one-dimensional substitution flows [A. I. Bufetov and B. Solomyak. A spectral cocycle for substitution systems and translation flows. J. Anal. Math.141(1) (2020), 165–205] is extended to the setting of pseudo-self-similar tilings in ${\mathbb R}^d$ , allowing expanding similarities with rotations. The pointwise upper Lyapunov exponent of this cocycle is used to bound the local dimension of spectral measures of deformed tilings. The deformations are considered, following the work of Treviño [Quantitative weak mixing for random substitution tilings. Israel J. Math., to appear], in the simpler, non-random setting. We review some of the results of Treviño in this special case and illustrate them on concrete examples.

Asymptotics of noncolliding q-exchangeable random walks

We consider a process of noncolliding q-exchangeable random walks on making steps 0 (‘straight’) and −1 (‘down’). A single random walk is called q-exchangeable if under an elementary transposition of the neighboring steps the probability of the trajectory is multiplied by a parameter . Our process of m noncolliding q-exchangeable random walks is obtained from the independent q-exchangeable walks via the Doob’s h-transform for a nonnegative eigenfunction h (expressed via the q-Vandermonde product) with the eigenvalue less than 1. The system of m walks evolves in the presence of an absorbing wall at 0. The repulsion mechanism is the q-analogue of the Coulomb repulsion of random matrix eigenvalues undergoing Dyson Brownian motion. However, in our model, the particles are confined to the positive half-line and do not spread as Brownian motions or simple random walks. We show that the trajectory of the noncolliding q-exchangeable walks started from an arbitrary initial configuration forms a determinantal point process, and express its kernel in a double contour integral form. This kernel is obtained as a limit from the correlation kernel of q-distributed random lozenge tilings of sawtooth polygons. In the limit as , with γ > 0 fixed, and under a suitable scaling of the initial data, we obtain a limit shape of our noncolliding walks and also show that their local statistics are governed by the incomplete beta kernel. The latter is a distinguished translation invariant ergodic extension of the two-dimensional discrete sine kernel.

Multi-focusing surface-emitting lasers.

Complete control of a beam pattern requires not only projecting a two-dimensional (2D) pattern but also focusing on a three-dimensional (3D) point cloud, which is typically achieved utilizing holography under the framework of diffraction. We previously reported direct focusing from on-chip size surface-emitting lasers that utilize a holographically modulated photonic crystal cavity based on 3D holography. However, this demonstration was of the simplest 3D hologram with a single point and single focal length, and the more typical 3D hologram with multiple points and multiple focal lengths has not yet been examined. Toward direct generation of a 3D hologram from the on-chip size surface-emitting laser, we here examined a simple 3D hologram featuring two different focal lengths with a single off-axis point in each to reveal the fundamental physics. Two types of holography, one based on superimposition and the other on random tiling, successfully demonstrated the desired focusing profiles. However, both types caused a spot noise beam in the far field plane due to interference between focusing beams with different focal lengths, especially in the case of the superimposing method. We also found that the 3D hologram based on the superimposing method consisted of higher order beams including the original hologram due to the manner of the holography. Secondly, we demonstrated a typical 3D hologram with multiple points and focal lengths and successfully showed the desired focusing profiles by both methods. We believe our findings will bring innovation to mobile optical systems and pave the way to developing compact optical systems in areas such as material processing, micro fluidics, optical tweezers, and endoscopy.

Deltoid Tangents with Evenly Distributed Orientations and Random Tilings

We study the construction of substitution tilings of the plane based on certain simplicial configurations of tangents of the deltoid with evenly distributed orientations. The random tiling ensembles are obtained as a result of tile rearrangements in the substitution rules associated to edge flips. Special types of random tilings for Euclidean, spherical and hyperbolic three-manifolds are also considered.

A columnar liquid quasicrystal with a honeycomb structure that consists of triangular, square and trapezoidal cells.

Quasicrystals are intriguing structures that have long-range positional correlations but no periodicity in real space, and typically with rotational symmetries that are 'forbidden' in conventional periodic crystals. Here, we present a two-dimensional columnar liquid quasicrystal with dodecagonal symmetry. Unlike previous dodecagonal quasicrystals based on random tiling, a honeycomb structure based on a strictly quasiperiodic tessellation of tiles is observed. The structure consists of dodecagonal clusters made up of triangular, square and trapezoidal cells that are optimal for local packing. To maximize the presence of such dodecagonal clusters, the system abandons periodicity but adopts a quasiperiodic structure that follows strict packing rules. The stability of random-tiling dodecagonal quasicrystals is often attributed to the entropy of disordering when strict tiling rules are broken, at the sacrifice of the long-range positional order. However, our results demonstrate that quasicrystal stability may rest on energy minimization alone, or with only minimal entropic intervention.

Measurable Tilings by Abelian Group Actions

Abstract Let $X$ be a measure space with a measure-preserving action $(g,x) \mapsto g \cdot x$ of an abelian group $G$. We consider the problem of understanding the structure of measurable tilings $F \odot A = X$ of $X$ by a measurable tile $A \subset X$ translated by a finite set $F \subset G$ of shifts, thus the translates $f \cdot A$, $f \in F$ partition $X$ up to null sets. Adapting arguments from previous literature, we establish a “dilation lemma” that asserts, roughly speaking, that $F \odot A = X$ implies $F^{r} \odot A = X$ for a large family of integer dilations $r$, and use this to establish a structure theorem for such tilings analogous to that established recently by the second and fourth authors. As applications of this theorem, we completely classify those random tilings of finitely generated abelian groups that are “factors of iid”, and show that measurable tilings of a torus ${\mathbb{T}}^{d}$ can always be continuously (in fact linearly) deformed into a tiling with rational shifts, with particularly strong results in the low-dimensional cases $d=1,2$ (in particular resolving a conjecture of Conley, the first author, and Pikhurko in the $d=1$ case).

Double interlacing in random tiling models

Random tilings of very large domains will typically lead to a solid, a liquid, and a gas phase. In the two-phase case, the solid–liquid boundary (arctic curve) is smooth, possibly with singularities. At the point of tangency of the arctic curve with the domain boundary, for large-sized domains, the tiles of a certain shape form a singly interlacing set, fluctuating according to the eigenvalues of the principal minors of a Gaussian unitary ensemble-matrix. Introducing non-convexities in large domains may lead to the appearance of several interacting liquid regions: They can merely touch, leading to either a split tacnode (hard tacnode), with two distinct adjacent frozen phases descending into the tacnode, or a soft tacnode. For appropriate scaling of the non-convex domains and probing about such split tacnodes, filaments, evolving in a bricklike sea of dimers of another type, will connect the liquid patches. Nearby, the tiling fluctuations are governed by a discrete tacnode kernel—i.e., a determinantal point process on a doubly interlacing set of dots belonging to a discrete array of parallel lines. This kernel enables us to compute the joint distribution of the dots along those lines. This kernel appears in two very different models: (i) domino tilings of skew-Aztec rectangles and (ii) lozenge tilings of hexagons with cuts along opposite edges. Soft tacnodes appear when two arctic curves gently touch each other amid a bricklike sea of dimers of one type, unlike the split tacnode. We hope that this largely expository paper will provide a view on the subject and be accessible to a wider audience.

Biased 2 times 2 periodic Aztec diamond and an elliptic curve

We study random domino tilings of the Aztec diamond with a biased 2 times 2 periodic weight function and associate a linear flow on an elliptic curve to this model. Our main result is a double integral formula for the correlation kernel, in which the integrand is expressed in terms of this flow. For special choices of parameters the flow is periodic, and this allows us to perform a saddle point analysis for the correlation kernel. In these cases we compute the local correlations in the smooth disordered (or gaseous) region. The special example in which the flow has period six is worked out in more detail, and we show that in that case the boundary of the rough disordered region is an algebraic curve of degree eight.

Gaussian unitary ensemble in random lozenge tilings

This paper establishes a universality result for scaling limits of uniformly random lozenge tilings of large domains. We prove that whenever the boundary of the domain has three adjacent straight segments inclined under 120 degrees (measured in the direction internal to the domain) to each other, the asymptotics of tilings near the middle segment is described by the GUE–corners process of random matrix theory. An important step in our argument is to show that fluctuations of the height function of random tilings on essentially arbitrary simply-connected domains of diameter N have magnitude smaller than \(N^{1/2}\).

Six-Vertex Model as a Grassmann Integral, One-Point Function, and the Arctic Ellipse

We formulate the six-vertex model with domain wall boundary conditions in terms of an integral over Grassmann variables. Relying on this formulation, we propose a method of calculation of correlation functions of the model for the case of the weights satisfying the free-fermion condition. We consider here in details the one-point correlation function describing the probability of a given state on an arbitrary edge of the lattice. We show that in the thermodynamic limit performed in such a way that the lattice is scaled to the square of unit side length, this function exhibits the “arctic ellipse” phenomenon, in agreement with the previous studies on random domino tilings of Aztec diamonds: it approaches its limiting values outside of an ellipse inscribed into this square, and takes continuously intermediate values inside the ellipse. We derive also the scaling properties of the one-point function in the vicinity of an arbitrary point of the arctic ellipse and in the vicinities of the points where the ellipse touches the boundary.

Local geometry of the rough-smooth interface in the two-periodic Aztec diamond

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 dominoes decay exponentially. In a previous paper, the authors found that a certain averaging of height function differences at the rough-smooth interface converged to the extended Airy kernel point process. In this paper, we augment the local geometrical picture at this interface by introducing well-defined lattice paths which are closely related to the level lines of the height function. We show, after suitable centering and rescaling, that a point process from these paths converge to the extended Airy kernel point process provided that the natural parameter associated to the two-periodic Aztec diamond is small enough.

The Largest Quasicrystalline Tiling with Dodecagonal Symmetry from a Single Pentablock Quarterpolymer of the AB1CB2D Type.

A quasicrystalline tiling pattern with tile size of ca. 60 nm has been discovered in the bulk state of a four-component pentablock polymer molecule of the AS1IS2P type, where A, S, I, and P denote poly(4-vinylbenzyldimethylamine), polystyrene, polyisoprene, and poly(2-vinylpyridine), respectively. The polymer samples used were prepared by anionic polymerizations and have narrow molecular weight distribution. The sample films were obtained by an extremely slow solvent-cast process from dilute solutions of tetrahydrofuran for 14 days. It has been found by TEM observation that the quarterpolymer, AS1IS2P-4 (Mn = 149 kg/mol, ϕA/ϕS1/ϕI/ϕS2/ϕP = 0.12/0.27/0.20/0.29/0.13), reveals two final stable structures, i.e., a 3.3.4.3.4 periodic tiling pattern as a minor component and a quasicrystalline (QC) tiling with dodecagonal symmetry as a major component, where the former includes a triangle/square number ratio of 2 and the latter has one of approximately 2.28, which is close enough to the ideal ratio, 4/ 2.31, for the triangle/square random tiling of the dodecagonal QC tiling (DDQC). Two structures were also clearly proved by SAXS diffraction patterns. Here, it should be noted this QC structure having a tile side length of ca. 60 nm was created with a single block polymer molecule.

Flow-induced-crystallization: tailoring host-guest supramolecular co-assemblies at the liquid-solid interface.

Here, we report that using the method of simply contacting a sample solution droplet with a piece of tissue paper can create a solvent flow (capillary force). During this process, the dynamics and solvent removal can promote the formation and stabilization of a meta-stable linear quasi-crystal composed of p-terphenyl-3,5,3′,5′-tetracarboxylic acid (TPTC) molecules, which would otherwise pack into thermodynamically favored random tiling. The tailored quasi-crystal (linear) template allows atop it higher-efficiency accommodation of fullerene molecules (C60) from 40.1% to 97.5%, compared with that obtained in the random-tiling (porous) case. Overall, the result of this study presents an unusual yet remarkably simple strategy for tailoring complex host–guest supramolecular systems at the liquid–solid interface.

ASYMPTOTIC EXPRESSION FOR RAW MOMENT OF THE MULTIPLICITY OF A PART IN A GENERAL PARTITION FUNCTION

Partitions appear across physics like Bose-Einstein statistics, dimer coverings, Potts model , crystal growth, Vasiliev theory, quasicrystals , Penrose-like random tilings, and so on. Since the general analytical study of partitions is rather tough, so statistical study is immensely preferred. This paper derives an asymptotic formula to evaluate the m -th raw moment of the multiplicity of a part for a broad category of partitions given by a specific generating function. In addition to being a multifold refinement/generalization of results in the literature, this work allows to calculate useful statistical quantities like mean and variance of the number of appearance of a part which is of deeper significance to analysts. The obtained expression is numerically validated to be in close agreement with the actual values. This study instigates the development of a general paradigm where partition theory is exposed to more sophisticated statistical analysis, thus, opening new avenues of research in this field.