Random Lattice Research Articles

Disordered hyperuniformity in superconducting vortex lattices

Particles occupying sites of a random lattice present density fluctuations at all length scales. It has been proposed that increasing interparticle interactions reduces long range density fluctuations, deviating from random behaviour. This leads to power laws in the structure factor and the number variance that can be used to characterize deviations from randomness which eventually lead to disordered hyperuniformity. It is not yet fully clear how to link density fluctuations with interactions in a disordered hyperuniform system. Interactions between superconducting vortices are very sensitive to vortex pinning, to the crystal structure of the superconductor and to the value of the magnetic field. This creates lattices with different degrees of disorder. Here we study disordered vortex lattices in several superconducting compounds (Co-doped NbSe$_2$, LiFeAs and CaKFe$_4$As$_4$) and in two amorphous W-based thin films, one with strong nanostructured pinning (W-film-1) and another one with weak or nearly absent pinning (W-film-2). We calculate for each case the structure factor and number variance and compare to calculations on an interacting set of partially pinned particles. We find that random density fluctuations appear when pinning overcomes interactions and show that the suppression of density fluctuations is indeed correlated to the presence of interactions. Furthermore, we find that we can describe all studied vortex lattices within a single framework consisting of a continous deviation from hyperuniformity towards random distributions when increasing the strength of pinning with respect to the intervortex interaction.

A classical density functional theory model for fragility in the hard-sphere limit

Abstract We study, using the classical density functional theory (DFT), the fragility and short-time elastic constants of a soft-sphere liquid. For the amorphous state, the order parameter is the inhomogeneous density function $\rho({\bf r})$ which is described in terms of Gaussian density profiles centered on a set random lattice points $\{{\bf R}_i\}$. The latter is characterized in terms of the Bernel pair function $g_\mathrm{B}(r)$. Based on the Adam–Gibbs-type relation between the $\alpha$ relaxation time $\tau_\alpha$ and the configurational entropy $\mathcal{S}_{\rm c}$, a thermodynamic fragility $m_\mathrm{T}$ for the liquid is defined. The concentration or average density of the liquid is treated as the control parameter here instead of temperature. The configurational entropy of the liquid is calculated using the DFT model. Variations in the short-range structure of the amorphous state are made with different choices for the value of $g_\mathrm{B}(r)$ at short distances, and its implications on the correlation between fragility $m_\mathrm{T}$ and the softness index $n$ are studied. The dependence of Poisson’s ratio $\nu$ on the softness index $n$ of the interaction potential is also obtained from the density dependence of the metastable state free energy. The correlation between $m_\mathrm{T}$ and $\nu$ follows.

Hipster random walks

We introduce and study a family of random processes on trees we call hipster random walks, special instances of which we heuristically connect to the min-plus binary trees introduced by Robin Pemantle and studied by Auffinger and Cable (Pemantle’s Min-Plus Binary Tree, 2017. arXiv:1709.07849 [math.PR]), and to the critical random hierarchical lattice studied by Hambly and Jordan (Adv Appl Probab 36(3):824–838, 2004. https://doi.org/10.1239/aap/1093962236 ). We prove distributional convergence for the processes, after rescaling, by showing that their evolutions can be understood as a discrete analogues of certain convection–diffusion equations, then using a combination of coupling arguments and results from the numerical analysis literature on convergence of numerical approximations of PDEs.

Role of Random and Coincidence Site Lattice Grain Boundaries in Liquid Metal Embrittlement of Iron (FCC)-Zn Couple

Liquid metal embrittlement (LME) has frequently been reported in various systems: Cu-Bi, Ni-Bi, Al-Ga, and Fe-Zn for more than 60 years. Although advances have been made in understanding the phenomenon, the role of grain boundary (GB) type and characteristics in LME has remained unclear. The present work shows that liquid metal penetration only occurs in random GBs, where its propagation path is a function of misorientation angle (θ) and stress component perpendicular to GB plane. In contrast, low-Σ coincidence site lattice (CSL) boundaries: Σ3 (θ = 60 deg) and Σ5 to 9 (~ θ = 40 deg) resist LME, even at the maximum stress component. Triple junctions of CSL and random GB block liquid metal penetration by modifying of the random GB misorientation angle. These findings provide insights to employ grain boundary engineering techniques to increase population of CSLs over random GBs and, hence, mitigate LME.

Emergence of scale-free smectic rivers and critical depinning in emulsions driven through disorder

During the past 60 min, oil companies have extracted 6 trillion liters of oil from the ground, thereby giving a striking illustration of the impact of multiphase flows on the world economy. From a fundamental perspective, we largely understand the dynamics of interfaces separating immiscible fluids driven through heterogeneous environments. In stark contrast, the basic mechanisms ruling the transport of fragmented fluids, such as foams and emulsions, remain elusive with studies mostly limited to isolated droplets and bubbles. Here, we demonstrate that the mobilization of emulsion driven through model disordered media is a critical plastic depinning transition. To elucidate this collective dynamics, we track the trajectories of hundreds of thousands of microfluidic droplets advected through random lattices of pinning sites. Their dynamics reveals that macroscopic mobilization only requires the coordinated motion of small groups of particles and does not involve any large-scale avalanches. Criticality arises from the interplay between contact and hydrodynamic interaction, which channel seemingly erratic depinning events along smectic river networks correlated over system spanning scales. Beyond the specifics of emulsion transport, we close our article discussing the similarities and profound differences with the plastic depinning transitions of driven flux lines in high-T c superconductors, charged colloids, and grain transport in eroded sand beds.

Multifunctional twisted kagome lattices: Tuning by pruning mechanical metamaterials.

This article investigates phonons and elastic response in randomly diluted lattices constructed by combining (via the addition of next-nearest bonds) a twisted kagome lattice, with bulk modulus B=0 and shear modulus G>0, with either a generalized untwisted kagome lattice with B>0 and G>0 or with a honeycomb lattice with B>0 and G=0. These lattices exhibit jamming-like critical endpoints at which B,G, or both B and G jump discontinuously from zero while the remaining moduli (if any) begin to grow continuously from zero. Pairs of these jamming points are joined by lines of continuous rigidity percolation transitions at which both B and G begin to grow continuously from zero. The Poisson ratio and G/B can be continuously tuned throughout their physical range via random dilution in a manner analogous to "tuning by pruning" in random jammed lattices. These lattices can be produced with modern techniques, such as three-dimensional printing, for constructing metamaterials.

Construction of one-dimensional random walk lattices using DNA algorithmic self-assembly

Algorithmic DNA lattices are constructed using pre-defined rules such as COPY, NOT, and XOR, where patterns are predicted based on initial values. However, the experimental implementation of an unpredictable random walk pattern (which is the implementation of a random rule, i.e., equally probable to move toward either the left or right in 1D systems) in DNA has not been reported yet. Here, we construct DNA lattices with DNA rule tiles implemented using the random rule. Patterns are visualized by atomic force microscopy. Finally, we discussed the average displacement, mean-square displacement, and number of displacement occurrences of experimental as well as simulated 1D random walk. The encoded information in sticky ends of DNA rule tiles demonstrates the feasibility of universal computation through DNA algorithmic self-assembly, which could be extremely beneficial in future computations.

On the universality of the Epstein zeta function

We study universality properties of the Epstein zeta function E_n(L,s) for lattices L of large dimension n and suitable regions of complex numbers s . Our main result is that, as n\to\infty , E_n(L,s) is universal in the right half of the critical strip as L varies over all n -dimensional lattices L . The proof uses a novel combination of an approximation result for Dirichlet polynomials, a recent result on the distribution of lengths of lattice vectors in a random lattice of large dimension and a strong uniform estimate for the error term in the generalized circle problem. Using the same approach we also prove that, as n\to\infty , E_n(L_1,s)-E_n(L_2,s) is universal in the full half-plane to the right of the critical line as (L_1,L_2) varies over all pairs of n -dimensional lattices. Finally, we prove a more classical universality result for E_n(L,s) in the s -variable valid for almost all lattices L of dimension n . As part of the proof we obtain a strong bound of E_n(L,s) on the critical line that is subconvex for n\geq 5 and almost all n -dimensional lattices L .

Renormalization group flow of coupled tensorial group field theories: Towards the Ising model on random lattices

We introduce a new family of tensorial field theories by coupling different fields in a non-trivial way, with a view towards the investigation of the coupling between matter and gravity in the quantum regime. As a first step, we consider the simple case with two tensors of the same rank coupled together, with Dirac like kinetic kernel. We focus especially on rank-$3$ tensors, which lead to a power counting just-renormalizable model, and interpret Feynman graphs as Ising configurations on random lattices. We investigate the renormalization group flow for this model, using two different and complementary tools for approximations, namely, the effective vertex expansion method and finite-dimensional truncations for the flowing action. Due to the complicated structure of the resulting flow equations, we divided the work into two parts. In this first part we only investigate the fundamental aspects on the construction of the model and the different ways to get tractable renormalization group equations, while their numerical analysis will be addressed in a companion paper.

Tracer diffusion on a crowded random Manhattan lattice

We study, by extensive numerical simulations, the dynamics of a hard-core tracer particle (TP) in presence of two competing types of disorder—frozen convection flows on a square random Manhattan lattice and a crowded dynamical environment formed by a lattice gas of mobile hard-core particles. The latter perform lattice random walks, constrained by a single-occupancy condition of each lattice site, and are either insensitive to random flows (model A) or choose the jump directions as dictated by the local directionality of bonds of the random Manhattan lattice (model B). We focus on the TP disorder-averaged mean-squared displacement, (which shows a super-diffusive behaviour ∼t4/3, t being time, in all the cases studied here), on higher moments of the TP displacement, and on the probability distribution of the TP position X along the x-axis, for which we unveil a previously unknown behaviour. Indeed, our analysis evidences that in absence of the lattice gas particles the latter probability distribution has a Gaussian central part , where u = X/t2/3, and exhibits slower-than-Gaussian tails for sufficiently large t and u. Numerical data convincingly demonstrate that in presence of a crowded environment the central Gaussian part and non-Gaussian tails of the distribution persist for both models.

Roller nanoimprint lithography for low-cost nanoscale random lattice diffractive optical elements

In this paper, effective fabrication of high-resolution diffractive optical elements on a polymer substrate is demonstrated using roller nanoimprint lithography. A nanoscale diffraction grating mold, which can generate a random laser pattern, is fabricated by scan-and-repeat projection lithography. The mold is imprinted to a thin polymer on the soft substrate via roller nanoimprint lithography, which has high efficiency, high fidelity, and high throughput for mass production. This fabrication process can produce high-resolution nanostructures while reducing the cost substantially. We obtained large-area polymer diffractive optical elements with a flexible substrate, which can generate high-quality diffraction random lattice patterns with sub-250 nm resolution for an 808-nm wavelength laser. The diffractive optical elements have about 83.2% diffraction efficiency and 99.7% uniformity of random pattern intensity. It is believed that this fabrication technique can promote practical applications of diffractive optical elements, such as laser wavefront correction, face and activity recognition, and optical communication.

Vortices in Bose–Einstein Condensates with Random Depth Optical Lattice

Effects of random depth optical lattice (OL) on vortices in Bose–Einstein condensates (BECs) are studied. By numerically solving Gross–Pitaevskii equation for rotating BECs, we analyzed the property of a single vortex (slow rotating) and vortex lattice structures (fast rotating) in the presence of random depth optical lattice. It is shown that similar to speckle disorder and bichromatic optical lattice pseudorandom potential, random depth optical lattice can also localize a singe vortex. Random depth OL can melt square vortex lattice, while speckle or bichromatic OL disorder can melt triangular vortex lattice. We also show that random depth OL disorder can induce larger angular momentum.

Random beam lattice modeling method for catastrophic crack growth simulation of brittle-like materials

A random and irregular beam lattice modeling method is developed to predict the crack growth process of brittle-like materials. The model representing the random degree of beam lattice model is defined, which is used to study the influence of lattice mesh on the crack path prediction results. In the simulation, the coupling interactionsamong cracks initiation and growth can be considered. Crack growth process of two double-edge notched specimens, respectively, under shear and tension and direct-tension were simulated and compared with the corresponding experimental results. By comparison of simulation and experimental results, effectiveness of the developed beam lattice modeling method is verified, and can be used to simulate crack growth for studying the fracture mechanisms of brittle-like materials. And the regular beam lattice model can affect the crack paths prediction, and random and irregular beam lattice model is more suitable for simulating crack growth.

2D Non-adjacent coupled map lattice with q and its applications in image encryption

In this paper, a novel high dimensional spatiotemporal chaotic system based on 2D non-adjacent coupled map lattice (2DNACML) model is proposed. Each local lattice is influenced by other random lattices in different dimensions and the local map is defined as a fractional-like style. From the theoretical analysis and numerical simulation, it is found that ranges of system parameters are expanded and unfixed in the novel 2DNACML model. As a result, the changed system is improved and been applied in the image encryption with a mixed scrambling scheme with chaotic method and knights tour method and an efficient diffusion scheme for different images to show the effectiveness of the proposed system.

Hotspots Analysis Using Cyber-Physical-Social System for a Smart City

Internet of things plays a vital role in providing various services to users. Significant volumes of data are generated from the communication between a large numbers of heterogeneous devices over the Internet. Big data technology is generally used to handle the large volume of data. Complex networks are graphs (networks) having non-trivial topological features, such as random graphs and lattices. Big data of complex networks concerns big data methods that can be used to analyze massive structural data sets, including considerably large networks and sets of graphs. This study is based on the critical phenomenon arising in complex networks that enable us to analytically predict the hotspots in smart cities. Hotspots are places with significantly high communication traffic relative to others. In this study, we propose a cyber-physical-social system for the analysis of high communication traffic hotspots using telecom data. The proposed model constructs a graph, and perform social network analysis on it. The process of hotspot extraction is performed, followed by social network analysis, which is conducted by quantifying the importance of each hotspot based on network metrics. These metrics aid in determining the importance of each hotspot in a telecom data network. Our objective is to prioritize different areas and detect hotspots quickly. Our results indicate that the proposed model has an efficiency comparable with that of state of the art methods. This research study will be helpful for urban planning and development, as well as in upgrading telecommunication infrastructure.

Characterizations of random walks on random lattices and their ramifications

We analyze the behavior of a particle moving along a d-dimensional lattice and representing a “doubly stochastic” random walk. Unlike the classical random walk, the lattice is randomly generated upon particle’s landing at a node. At any time, the particle is enclosed in a rectangular cylinder (i.e., with a bounded a-dimensional rectangle Ra at its base) it attempts to escape. Thus, the particle moves from one node to another at random epochs of time and, with some probability, leaves The particle may jump arbitrarily far beyond the boundary at its passage time. Furthermore, the particle’s location is not assumed to be known in real time, but only upon certain random epochs Of a key interest is the location of the particle at any time, where is the linear interpolation of particle’s positions at times τn’s. If denotes the first observed escape time (virtual first passage time), where we target the joint characteristic function of and itself, where or in tractable forms, thereby attempting to enhance as much as possible the probabilistic data lost due to the crudeness of the observations and to couple and with deterministic time intervals Among various applications, we discuss and treat antagonistic games of two active and several passive players as well as situations that occur in complex queueing systems.

Alloying effects on low‒energy recoil events in concentrated solid‒solution alloys

Alloying elements at low-concentrations into pure metals is usually adopted to improve their mechanical properties and irradiation performance. Recently developed single-phase concentrated solid-solution alloys (CSAs) comprised of two or more elements, all at high concentrations, have demonstrated good radiation resistance. CSAs are characterized by their extreme disordered states that arise from the random arrangement of different elements and accompanied random local lattice distortions. In this work, we investigate low-energy recoil events in Ni0.5X0.5 (X = Fe and Co) CSAs and Ni0.8X0.2 (X = Fe, Co, Cr, and Pd) CSAs using ab initio molecular dynamics simulations to understand the effects of different disorder on defect production in CSAs. The threshold displacement energies are determined along three high-symmetric directions by randomly choosing independent primary knock-on atoms in each direction. As expected, the threshold energies in Ni and its CSAs are anisotropic, with the highest values found in the [111] direction. The calculated threshold energies of Fe in NiFe are smaller than those in NiCo and Ni, especially along the [111] direction. An inspection of the atomic trajectories inside the collision cascade reveals that the effect of chemical disorder outweighs the site-to-site lattice distortions in determining the threshold energies. Especially, different interaction properties between elements due to their different electronic structures are responsible for the observed different threshold energies. The local environment dependence of threshold energies suggests that local elemental arrangement can be used to understand and predict threshold energies in disordered alloys.

Epidemic outbreaks on random Voronoi–Delaunay triangulations

We study epidemic outbreaks on random Delaunay triangulations by applying the Asynchronous SIR (susceptible–infected–removed) dynamics coupled to two-dimensional Voronoi–Delaunay triangulations. In order to investigate the critical behavior of the model, we obtain the cluster size distribution by using Newman–Ziff algorithm, allowing to simulate random inhomogeneous lattices and measure any desired observable related to percolation. We numerically calculate the order parameter, defined as the wrapping cluster density, the mean cluster size, and Binder cumulant ratio defined for percolation in order to estimate the epidemic threshold. Our findings suggest that the system falls into two-dimensional dynamic percolation universality class and the quenched random disorder is irrelevant, in agreement with results for classical percolation.

Random sequential adsorption on Euclidean, fractal, and random lattices.

Irreversible adsorption of objects of different shapes and sizes on Euclidean, fractal, and random lattices is studied. The adsorption process is modeled by using random sequential adsorption algorithm. Objects are adsorbed on one-, two-, and three-dimensional Euclidean lattices, on Sierpinski carpets having dimension d between 1 and 2, and on Erdős-Rényi random graphs. The number of sites is M=L^{d} for Euclidean and fractal lattices, where L is a characteristic length of the system. In the case of random graphs, such a characteristic length does not exist, and the substrate can be characterized by a fixed set of M vertices (sites) and an average connectivity (or degree) g. This paper concentrates on measuring (i) the probability W_{L(M)}(θ) that a lattice composed of L^{d}(M) elements reaches a coverage θ and (ii) the exponent ν_{j} characterizing the so-called jamming transition. The results obtained for Euclidean, fractal, and random lattices indicate that the quantities derived from the jamming probability W_{L(M)}(θ), such as (dW_{L}/dθ)_{max} and the inverse of the standard deviation Δ_{L}, behave asymptotically as M^{1/2}. In the case of Euclidean and fractal lattices, where L and d can be defined, the asymptotic behavior can be written as M^{1/2}=L^{d/2}=L^{1/ν_{j}}, with ν_{j}=2/d.

Toward Realistic Amorphous Topological Insulators.

The topological properties of materials are, until now, associated with the features of their crystalline structure, although translational symmetry is not an explicit requirement of the topological phases. Recent studies of hopping models on random lattices have demonstrated that amorphous model systems show a nontrivial topology. Using ab initio calculations, we show that two-dimensional amorphous materials can also display topological insulator properties. More specifically, we present a realistic state-of-the-art study of the electronic and transport properties of amorphous bismuthene systems, showing that these materials are topological insulators. These systems are characterized by the topological index [Formula: see text]2 = 1 and bulk-edge duality, and their linear conductance is quantized, [Formula: see text], for Fermi energies within the topological gap. Our study opens the path to the experimental and theoretical investigation of amorphous topological insulator materials.