Random Lattice Research Articles

Concavity analysis of the tangent method

The tangent method has recently been devised by Colomo and Sportiello (2016 J. Stat. Phys. 164 1488–523) as an efficient way to determine the shape of arctic curves. Largely conjectural, it has been tested successfully in a variety of models. However no proof and no general geometric insight have been given so far, either to show its validity or to allow for an understanding of why the method actually works. In this paper, we propose a universal framework which accounts for the tangency part of the tangent method, whenever a formulation in terms of directed lattice paths is available. Our analysis shows that the key factor responsible for the tangency property is the concavity of the entropy (also called the Lagrangean function) of long random lattice paths. We extend the proof of the tangency to q-deformed paths.

Application of Convolutional Neural Network to Quantum Percolation in Topological Insulators

Quantum material phases such as the Anderson insulator, diffusive metal, and Weyl/Dirac semimetal as well as topological insulators show specific wave functions both in real and Fourier spaces. These features are well captured by convolutional neural networks, and the phase diagrams have been obtained, where standard methods are not applicable. One of these examples is the cases of random lattices such as quantum percolation. Here, we study the topological insulators with random vacancies, namely, the quantum percolation in topological insulators, by analyzing the wave functions via a convolutional neural network. The vacancies in topological insulators are especially interesting since peculiar bound states are formed around the vacancies. We show that only a few percent of vacancies are required for a topological phase transition. The results are confirmed by independent calculations of localization length, density of states, and wave packet dynamics.

Remote control of microtubule plus-end dynamics and function from the minus-end.

In eukaryotes, the organization and function of the microtubule cytoskeleton depend on the allocation of different roles to individual microtubules. For example, many asymmetrically dividing cells differentially specify microtubule behavior at old and new centrosomes. Here we show that yeast spindle pole bodies (SPBs, yeast centrosomes) differentially control the plus-end dynamics and cargoes of their astral microtubules, remotely from the minus-end. The old SPB recruits the kinesin motor protein Kip2, which then translocates to the plus-end of the emanating microtubules, promotes their extension and delivers dynein into the bud. Kip2 recruitment at the SPB depends on Bub2 and Bfa1, and phosphorylation of cytoplasmic Kip2 prevents random lattice binding. Releasing Kip2 of its control by SPBs equalizes its distribution, the length of microtubules and dynein distribution between the mother cell and its bud. These observations reveal that microtubule organizing centers use minus to plus-end directed remote control to individualize microtubule function.

Random lattices, punctured tori and the Teichmüller distribution

The moduli space of lattices of C \mathbb {C} is a Riemann surface of finite hyperbolic area with the square lattice as an origin. We select a lattice from the induced uniform distribution and calculate the statistics of the Teichmüller distance to the origin. This in turn identifies distribution of the distance in Teichmüller space to the central “square” punctured torus in the moduli space of punctured tori. There are singularities in this p.d.f. arising from the topology of the moduli space. We also consider the statistics of the distance in Teichmüller space to the rectangular punctured tori and the p.d.f. and expected distortion of the extremal quasiconformal mappings.

Network analysis of whole-brain fMRI dynamics: A new framework based on dynamic communicability

Neuroimaging techniques such as MRI have been widely used to explore the associations between brain areas. Structural connectivity (SC) captures the anatomical pathways across the brain and functional connectivity (FC) measures the correlation between the activity of brain regions. These connectivity measures have been much studied using network theory in order to uncover the distributed organization of brain structures, in particular FC for task-specific brain communication. However, the application of network theory to study FC matrices is often “static” despite the dynamic nature of time series obtained from fMRI. The present study aims to overcome this limitation by introducing a network-oriented analysis applied to whole-brain effective connectivity (EC) useful to interpret the brain dynamics. Technically, we tune a multivariate Ornstein-Uhlenbeck (MOU) process to reproduce the statistics of the whole-brain resting-state fMRI signals, which provides estimates for MOU-EC as well as input properties (similar to local excitabilities). The network analysis is then based on the Green function (or network impulse response) that describes the interactions between nodes across time for the estimated dynamics. This model-based approach provides time-dependent graph-like descriptor, named communicability, that characterize the roles that either nodes or connections play in the propagation of activity within the network. They can be used at both global and local levels, and also enables the comparison of estimates from real data with surrogates (e.g. random network or ring lattice). In contrast to classical graph approaches to study SC or FC, our framework stresses the importance of taking the temporal aspect of fMRI signals into account. Our results show a merging of functional communities over time, moving from segregated to global integration of the network activity. Our formalism sets a solid ground for the analysis and interpretation of fMRI data, including task-evoked activity.

A new method of generating hard random lattices with short bases

This paper first gives a regularity theorem and its corollary. Then, a new construction of generating hard random lattices with short bases is obtained by using this corollary. This construction is from a new perspective and uses a random matrix whose entries obeyed Gaussian sampling which ensures that the corresponding schemes have a wider application future in cryptography area. Moreover, this construction is more specific than the previous constructions, which makes it can be implemented easier in practical applications.

Influence of antidots on the critical current density of HTSC in a magnetic field

By using the Monte Carlo method, a numerical study has been conducted to calculate the current-voltage characteristics of a high-temperature superconductor with intrinsic pinning sites and the additional ones in the form of antidots in different magnetic fields. Varioustypes of antidot distribution among the sample such as a random distribution, triangular and square lattice and their conformal transformations, have been considered. Descending critical current density dependencies on the magnetic field for different samples have been acquired. Averaged vortex configurations of samples for different antidot distributions at the same magnetic field and transport current have been acquired. The most effective antidot distribution has been determined.

Novel Identity-Based Hash Proof System with Compact Master Public Key from Lattices in the Standard Model

As the progress of quantum computers, it is desired to propose many more efficient cryptographic constructions with post-quantum security. In the literatures, almost all cryptographic schemes and protocols can be explained and constructed modularly from certain cryptographic primitives, among which an Identity-Based Hash Proof System (IB-HPS) is one of the most basic and important primitives. Therefore, we can utilize IB-HPSs with post-quantum security to present several types of post-quantum secure schemes and protocols. Up until now, all known IB-HPSs with post-quantum security are instantiated based on latticed-based assumptions. However, all these lattice-based IB-HPSs are either in the random oracle model or not efficient enough in the standard model. Hence, it should be of great significance to construct more efficient IB-HPSs from lattices in the standard model. In this paper, we propose a new smooth IB-HPS with anonymity based on the Learning with Errors (LWE) assumption in the standard model. This new construction is mainly inspired by a classical identity-based encryption scheme based on LWE due to Agreawal et al. in Eurocrypt 2010. And our innovation is to employ the algorithm SampleGaussian introduced by Gentry et al. and the property of random lattice to simulate the identity secret key with respect to the challenge identity. Compared with other existing IB-HPSs in the standard model, our master public key is quite compact. As a result, our construction has much lower overheads on computation and storage.

Random lattice strain and its relaxation towards the morphotropic phase boundary of Na0.5Bi0.5TiO3 -based piezoelectrics: Impact on the structural and ferroelectric properties

We demonstrate that the lead-free piezoelectric compound Na0.5Bi0.5TiO3 (NBT) exhibits random lattice strain in the ferroelectric phase, and that this feature primarily dictates the way the system evolves towards the morphotropic phase boundary in the unpoled state of NBT-based piezoelectrics. Investigations on two different morphotropic phase boundary (MPB) systems, namely Na0.5Bi0.5TiO3 - K0.5Bi0.5TiO3 (NBT-KBT) and Na0.5Bi0.5TiO3 - BaTiO3 (NBT-BT), revealed that the coupled structural-polar evolution towards the MPB is primarily driven by the necessity to minimize this strain. Our study suggests that the random lattice strain originates in the random stacking of the in-phase tilt and antiphase octahedral tilted regions, and that the system is able to minimize it by adopting a sequential stacking of the two tilt types, leading to a long-period modulation in the octahedral tilt configuration over large parts of the sample volume. This hinders the development of long-range ferroelectric order as the MPB is approached. We also demonstrate that the composition showing the maximum piezoelectric coefficient corresponds to a structural state wherein considerable polar-structural disorder coexists with the field-stabilized long-range rhombohedral ferroelectric order after poling, and not coexistence of two ferroelectric phases (tetragonal and rhombohedral), generally believed.

Graph theoretical approaches for the characterization of damage in hierarchical materials

We discuss the relevance of methods of graph theory for the study of damage in simple model materials described by the random fuse model. While such methods are not commonly used when dealing with regular random lattices, which mimic disordered but statistically homogeneous materials, they become relevant in materials with microstructures that exhibit complex multi-scale patterns. We specifically address the case of hierarchical materials, whose failure, due to an uncommon fracture mode, is not well described in terms of either damage percolation or crack nucleation-and-growth. We show that in these systems, incipient failure is accompanied by an increase in eigenvector localization and a drop in topological dimension. We propose these two novel indicators as possible candidates to monitor a system in the approach to failure. As such, they provide alternatives to monitoring changes in the precursory avalanche activity, which is often invoked as a candidate for failure prediction in materials which exhibit critical-like behavior at failure, but may not work in the context of hierarchical materials which exhibit scale-free avalanche statistics even very far from the critical load.

Effect of TiB particles on the beta recrystallization behavior of the Ti-2Al-9.2Mo-2Fe-0.1B metastable beta titanium alloy

The effect of TiB particles on the β recrystallization behavior of an as-cast Ti-2Al-9.2Mo-2Fe-0.1B (hereafter 2A2F10B) metastable β titanium alloy was investigated during hot rolling with a related β solution treatment. The microstructure evolution was analyzed based on the electron backscatter diffraction (EBSD). The particle-stimulated nucleation (PSN) behavior of the TiB particles occurred naturally in the dynamic recrystallization during hot rolling and in the static recrystallization during the β solution treatment, which effectively promoted the recrystallization of 2A2F10B under the studied conditions. The PSN mechanism caused recrystallized grains with different orientations to nucleate around single TiB particles by random lattice rotation, which led to a weak texture in the recrystallized microstructure. The complete recrystallization of the as-cast 2A2F10B alloy could be achieved, and it had a fine microstructure after single-heating hot rolling, with a reduction of at least 65%, along with the related β solution treatment.

Engineering of Chern insulators and circuits of topological edge states

Impurities embedded in electronic systems induce bound states which under certain circumstances can hybridize and lead to impurity bands. Doping of insulators with impurities has been identified as a promising route towards engineering electronic topological states of matter. In this paper we show how to realize tuneable Chern insulators starting from a three dimensional topological insulator whose surface is gapped and intentionally doped with magnetic impurities. The main advantage of the protocol is that it is robust, and in particular not very sensitive to the impurity configuration. We explicitly demonstrate this for a square lattice of impurities as well as a random lattice. In both cases we show that it is possible to change the Chern number of the system by one through manipulating its topological state. We also discuss how this can be used to engineer circuits of edge channels.

Calculation of \u03c0 and Classification of Self-avoiding Lattices via DNA Configuration

Numerical simulation (e.g. Monte Carlo simulation) is an efficient computational algorithm establishing an integral part in science to understand complex physical and biological phenomena related with stochastic problems. Aside from the typical numerical simulation applications, studies calculating numerical constants in mathematics, and estimation of growth behavior via a non-conventional self-assembly in connection with DNA nanotechnology, open a novel perspective to DNA related to computational physics. Here, a method to calculate the numerical value of π, and way to evaluate possible paths of self-avoiding walk with the aid of Monte Carlo simulation, are addressed. Additionally, experimentally obtained variation of the π as functions of DNA concentration and the total number of trials, and the behaviour of self-avoiding random DNA lattice growth evaluated through number of growth steps, are discussed. From observing experimental calculations of π (πexp) obtained by double crossover DNA lattices and DNA rings, fluctuation of πexp tends to decrease as either DNA concentration or the number of trials increases. Based upon experimental data of self-avoiding random lattices grown by the three-point star DNA motifs, various lattice configurations are examined and analyzed. This new kind of study inculcates a novel perspective for DNA nanostructures related to computational physics and provides clues to solve analytically intractable problems.

On the generalized circle problem for a random lattice in large dimension

In this note we study the error term Rn,L(x) in the generalized circle problem for a ball of volume x and a random lattice L of large dimension n. Our main result is the following functional central limit theorem: Fix an arbitrary function f:Z+→R+ satisfying limn→∞⁡f(n)=∞ and f(n)=Oε(eεn) for every ε>0. Then, the random functiont↦12f(n)Rn,L(tf(n)) on the interval [0,1] converges in distribution to one- dimensional Brownian motion as n→∞. The proof goes via convergence of moments, and for the computations we develop a new version of Rogers' mean value formula from [18]. For the individual kth moment of the variable (2f(n))−1/2Rn,L(f(n)) we prove convergence to the corresponding Gaussian moment more generally for functions f satisfying f(n)=O(ecn) for any fixed c∈(0,ck), where ck is a constant depending on k whose optimal value we determine.

A Century of Topological Coevolution of Complex Infrastructure Networks in an Alpine City

In this paper, we used complex network analysis approaches to investigate topological coevolution over a century for three different urban infrastructure networks. We applied network analyses to a unique time‐stamped network data set of an Alpine case study, representing the historical development of the town and its infrastructure over the past 108 years. The analyzed infrastructure includes the water distribution network (WDN), the urban drainage network (UDN), and the road network (RN). We use the dual representation of the network by using the Hierarchical Intersection Continuity Negotiation (HICN) approach, with pipes or roads as nodes and their intersections as edges. The functional topologies of the networks are analyzed based on the dual graphs, providing insights beyond a conventional graph (primal mapping) analysis. We observe that the RN, WDN, and UDN all exhibit heavy tailed node degree distributions [P(k)] with high dispersion around the mean. In 50 percent of the investigated networks, P(k) can be approximated with truncated [Pareto] power‐law functions, as they are known for scale‐free networks. Structural differences between the three evolving network types resulting from different functionalities and system states are reflected in the P(k) and other complex network metrics. Small‐world tendencies are identified by comparing the networks with their random and regular lattice network equivalents. Furthermore, we show the remapping of the dual network characteristics to the spatial map and the identification of criticalities among different network types through co‐location analysis and discuss possibilities for further applications.

Dual random lattice modeling of backward erosion piping

This manuscript provides a novel random lattice model to simulate the progressive degradation of soil embankments induced by the backward erosion piping (BEP) phenomenon. The progressive evolution of the piping process is described by expressing apparent diffusivity of the soil as a function of the local hydraulic gradient. In order to accurately compute local field gradients, a dual lattice approach that evaluates the response on both Delaunay and Voronoi nodes is formulated. The response computed at the Delaunay nodes is then employed to augment field gradient computation on the Voronoi grid. The proposed dual lattice model is numerically verified and validated by comparing numerical results with BEP experiments available in the literature.

Long term behavior of a random Hopfield neural lattice model

A Hopfield neural lattice model is developed as the infinite dimensional extension of the classical finite dimensional Hopfield model. In addition, random external inputs are considered to incorporate environmental noise. The resulting random lattice dynamical system is first formulated as a random ordinary differential equation on the space of square summable bi-infinite sequences. Then the existence and uniqueness of solutions, as well as long term dynamics of solutions are investigated.

The Gaussian core model in high dimensions

We prove lower bounds for energy in the Gaussian core model, in which point particles interact via a Gaussian potential. Under the potential function $t \mapsto e^{-\alpha t^2}$ with $0 < \alpha < 4\pi/e$, we show that no point configuration in $\mathbf{R}^n$ of density $\rho$ can have energy less than $(\rho+o(1))(\pi/\alpha)^{n/2}$ as $n \to \infty$ with $\alpha$ and $\rho$ fixed. This lower bound asymptotically matches the upper bound of $\rho (\pi/\alpha)^{n/2}$ obtained as the expectation in the Siegel mean value theorem, and it is attained by random lattices. The proof is based on the linear programming bound, and it uses an interpolation construction analogous to those used for the Beurling-Selberg extremal problem in analytic number theory. In the other direction, we prove that the upper bound of $\rho (\pi/\alpha)^{n/2}$ is no longer asymptotically sharp when $\alpha > \pi e$. As a consequence of our results, we obtain bounds in $\mathbf{R}^n$ for the minimal energy under inverse power laws $t \mapsto 1/t^{n+s}$ with $s>0$, and these bounds are sharp to within a constant factor as $n \to \infty$ with $s$ fixed.

Majority-vote model on directed Small-World-Voronoi–Delaunay random lattices

We investigate the critical properties of the nonequilibrium majority-vote model in two-dimensions on directed small-world lattice with quenched connectivity disorder. The disordered system is studied through Monte Carlo simulations: the critical noise ([Formula: see text]), as well as the critical exponents [Formula: see text], [Formula: see text], and [Formula: see text] for several values of the rewiring probability [Formula: see text]. We find that this disordered system does not belong to the same universality class as the regular two-dimensional ferromagnetic model. The majority-vote model on directed small-world lattices presents in fact a second-order phase transition with new critical exponents which do not depend on [Formula: see text] ([Formula: see text]), but agree with the exponents of the equilibrium Ising model on directed small-world Voronoi–Delaunay random lattices.

Equilibrium states for random lattice models of hyperbolic type

We study the structural stability of random coupled map lattice models of hyperbolic type under certain metrics. Furthermore, by describing the thermodynamic formalism of the underlying random spin lattice system, we prove the existence of equilibrium states for equi-Hölder continuous random functions on lattice models under the conditions of random weak interaction and translation invariance and present some partial results on the uniqueness of equilibrium states.