Random Lattice Research Articles

Nonuniversal critical dynamics on planar random lattices with heterogeneous degree distributions

The weighted planar stochastic (WPS) lattice introduces a topological disorder that emerges from a multifractal structure. Its dual network has a power-law degree distribution and is embedded in a two-dimensional space, forming a planar network. We modify the original recipe to construct WPS networks with degree distributions interpolating smoothly between the original power-law tail, P(q)∼q−α with exponent α≈5.6, and a square lattice. We analyze the role of the disorder in the modified WPS model, considering the critical behavior of the contact process (CP). We report a critical scaling depending on the network degree distribution. The scaling exponents differ from the standard mean-field behavior reported for CP on infinite-dimensional (random) graphs with power-law degree distribution. Furthermore, the disorder present in the WPS lattice model is in agreement with the Luck-Harris criterion for the relevance of disorder in critical dynamics. However, despite the same wandering exponent ω=1/2, the disorder effects observed for the WPS lattice are weaker than those found for uncorrelated disorder.

Designing Switchable Transition Metal Nitrides/Carbides from High-Entropy PBA Derivative As Bifunctional Oxygen Electrocatalyst for Rechargeable Zinc-Air Batteries

In response to the growing energy and environmental challenges, considerable endeavors have been directed towards the development of sustainable and eco-friendly energy technologies as alternatives to fossil fuels. Moreover, rechargeable zinc-air batteries (RZABs) stand out due to their high energy density and environmental friendliness, although the sluggish cathode oxygen reduction reaction (ORR) and oxygen evolution reaction (OER) kinetics restrict RZABs performance. Prussian blue analogues (PBAs), owing to their highly porous structure and tunable morphology, are often utilized as the precursors of the materials. Particularly, high-entropy PBAs (HEPBAs), formed by blending five or more metals in a random lattice, have achieved enhanced thermal and chemical stability via a robust structure with multiple transition metal ions. Meanwhile, transition metal carbides (TMCs) were investigated for ZAB applications due to their higher catalytic activity. Herein, we introduced CNT into HEPBA nanoparticles, followed by calcination processes, resulting in the synthesis of high-entropy metal carbides on CNT (HE-TMCs/CNT) which are effective for the bifunctional OER and ORR performance. HE-TMCs/CNT demonstrate superior activity towards the ORR (E1/2 = 0.77 V) in a 0.1 M KOH solution and OER (η = 330 mV @10 mA cm−2) in a 1 M KOH solution. The ZAB with HE-TMCs/CNT as the cathode demonstrated an open-circuit voltage of 1.39 V, providing a high energy density of 71 mW cm-2 under a current density of 102 mA cm-2. It exhibited the ability to be charged and discharged in cycles for up to 40 hours, revealing good stability under an applied current density of 5 mA cm-2. These results offer a straightforward pathway for the development of efficient bifunctional electrocatalysts for high-performance RZABs. Figure 1

Multi-index upper semicontinuity of pullback random attractors for fractional discrete FitzHugh-Nagumo systems with delays driven by Wong-Zakai approximations

In this paper, we consider the global well-posedness and multi-index stability of pullback random attractors for random fractional retarded lattice FitzHugh-Nagumo systems with nonlinear Wong-Zakai noise and non-autonomous forcing term. We use the idea of Caraballo et al. (2014) [2] to prove the global well-posedness. Based on the global well-posedness of the lattice FitzHugh-Nagumo system, we study the existence, uniqueness and upper semicontinuity of pullback random attractors Aϱδ={Aϱδ(τ,ω):τ∈R,ω∈Ω}. More precisely, we first utilize the method of tail-estimates of solution and Ascoli-Arzelà theorem to prove the existence and uniqueness of pullback random attractors, and then investigate the four types of upper semicontinuity of pullback random attractors: (1) The long time stability of pullback random attractors as the time parameter τ approaches negative infinity; (2) The upper semicontinuity of pullback random attractors as the step-length δ of the Wong-Zakai approximation tends to positive infinity; (3) The upper semicontinuity of pullback random attractors as the delay time ϱ→0; (4) The upper semicontinuity of pullback random attractors from non-autonomous to autonomous.

Critical Behavior of the Stochastic SIR Model on Random Bond-Diluted Lattices

Critical Behavior of the Stochastic SIR Model on Random Bond-Diluted Lattices

Quantum-Resistant Forward-Secure Digital Signature Scheme Based on q-ary Lattices

In this paper, we design and consider a new digital signature scheme with an evolving secret key, using random q-ary lattices as its domain. It is proved that, in addition to offering classic eu-cma security, the scheme is existentially forward unforgeable under an adaptive chosen message attack (fu-cma). We also prove that the secret keys are updated without revealing anything about any of the keys from the prior periods. Therefore, we design a polynomial-time reduction and use it to show that the ability to create a forgery leads to a feasible method of solving the well-known small integer solution (SIS) problem. Since the security of the scheme is based on computational hardness of a SIS problem, it turns out to be resistant to both classic and quantum methods. In addition, the scheme is based on the "Fiat-Shamir with aborts" approach that foils a transcript attack. As for the key-updating mechanism, it is based on selected properties of binary trees, with the number of leaves being the same as the number of time periods in the scheme. Forward security is gained under the assumption that one out of two hash functions is modeled as a random oracle.

Gas Sorption Characterization of Porous Materials Employing a Statistical Theory for Bethe Lattices.

In the present work, a recently developed statistical theory for adsorption and desorption processes in mesoporous solids, modeled by random Bethe lattices, has been applied to obtain pore size distributions and interpore connectivity from sorption isotherms in real random porous materials, employing a robust and validated methodology. Using the experimental adsorption-desorption N2 isotherms at 77.4 K on Vycor glass, a porous material with random pore structure, we demonstrate the solution of the inverse problem resulting in extracted pore size distribution and interpore connectivity, notably different from the predictions of earlier theories. The results presented are corroborated by the analysis of 3D digital images of reconstructed Vycor porous glass, showing excellent agreement between the predictions of geometric analysis and the new statistical theory.

Dynamics of a Random Hopfield Neural Lattice Model with Adaptive Synapses and Delayed Hebbian Learning

Dynamics of a Random Hopfield Neural Lattice Model with Adaptive Synapses and Delayed Hebbian Learning

Approximation of (some) random FPUT lattices by KdV equations

We consider a Fermi–Pasta–Ulam–Tsingou lattice with randomly varying coefficients. We discover a relatively simple condition which when placed on the nature of the randomness allows us to prove that small amplitude/long wavelength solutions are almost surely rigorously approximated by solutions of Korteweg–de Vries equations for very long times. The key ideas combine energy estimates with homogenization theory and the technical proof requires a novel application of autoregressive processes.

Vibrational spectrum of Granular packings with random matrices

The vibrational spectrum of granular packings can be used as a signature of the jamming transition, with the density of states at zero frequency becoming nonzero at the transition. It has been proposed previously that the vibrational spectrum of granular packings can be approximately obtained from random matrix theory. Here, we show that the autocorrelation function of the density of states shows good agreement between dynamical numerical simulations of frictionless bead packs near the jamming point and the analytic predictions of the Laguerre orthogonal ensemble of random matrices; there is clear disagreement with the Gaussian orthogonal ensemble, establishing that the Laguerre ensemble correctly reproduces the universal statistical properties of jammed granular matter and excluding the Gaussian orthogonal ensemble. We also present a random lattice model which is a physically motivated variant of the random matrix ensemble. Numerical calculations reveal that this model reproduces the known features of the vibrational density of states of frictionless granular matter, while also retaining the correlation structure seen in the Laguerre random matrix theory.Graphic abstract

Residually continuous pullback attractors for stochastic discrete plate equations

For a nonautonomous stochastic discrete plate equation driven by multiplicative noise, we prove the unique existence of a pullback random attractor, which is a family of pullback‐attracting random compact sets parameterized by time and samples. We then establish the residual dense continuity of the pullback random attractor on the time‐sample plane with respect to the Hausdorff metric. Even without the Cantor continuum hypothesis, we show that the set of all points of continuity of the pullback random attractor is an uncountable set with the continuity cardinality. These results explain both nonexplosive and nonimplosive phenomena for the random plate lattice system.

Dynamics of a random Hopfield neural lattice model with adaptive synapses and delayed Hebbian learning

UDC 517.9 A Dong–Hopfield neural lattice model with random external forcing and delayed response to the evolution of interconnection weights is developed and studied. The interconnection weights evolve according to the Hebbian learning rule with a decay term and contribute to changes in the states after a short delay. The lattice system is first reformulated as a coupled functional-ordinary differential equation system on an appropriate product space. Then the solution of the system is shown to exist and be unique. Furthermore it is shown that the system of equations generates a continuous random dynamical system. Finally, the existence of random attractors for the random dynamical system generated by the Dong–Hopfield model is established.

Upper Bound for the Coefficients of the Shortest Vector of Random Lattice

This paper shows that upper bounds on the coefficients of the shortest vector of a lattice can be represented using the smallest eigenvalue of the Gram matrix for the lattice, obtains its distribution for high-dimensional random Goldstein-Mayer lattice, and applies it to determine the percentage of zeros of coefficient vector.

An exact chiral amorphous spin liquid

Topological insulator phases of non-interacting particles have been generalized from periodic crystals to amorphous lattices, which raises the question whether topologically ordered quantum many-body phases may similarly exist in amorphous systems? Here we construct a soluble chiral amorphous quantum spin liquid by extending the Kitaev honeycomb model to random lattices with fixed coordination number three. The model retains its exact solubility but the presence of plaquettes with an odd number of sides leads to a spontaneous breaking of time reversal symmetry. We unearth a rich phase diagram displaying Abelian as well as a non-Abelian quantum spin liquid phases with a remarkably simple ground state flux pattern. Furthermore, we show that the system undergoes a finite-temperature phase transition to a conducting thermal metal state and discuss possible experimental realisations.

Smoothing Codes and Lattices: Systematic Study and New Bounds

In this article we revisit smoothing bounds in parallel between lattices <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">and</i> codes. Initially introduced by Micciancio and Regev, these bounds were instantiated with Gaussian distributions and were crucial for arguing the security of many lattice-based cryptosystems. Unencumbered by direct application concerns, we provide a systematic study of how these bounds are obtained for both lattices <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">and</i> codes, transferring techniques between both areas. We also consider multiple choices of spherically symmetric noise distributions. We found that the best strategy for a worst-case bound combines Parseval’s Identity, the Cauchy-Schwarz inequality, and the second linear programming bound, and this holds for both codes and lattices and all noise distributions at hand. For an average-case analysis, the linear programming bound can be replaced by an expected value computation. This alone gives optimal results for spherically uniform noise over random codes and random lattices. This also improves prior Gaussian smoothing bounds for worst-case lattices, but surprisingly this provides even better results with uniform ball noise than for Gaussian (or Bernoulli noise for codes). This counterintuitive situation can be resolved by adequate decomposition and truncation of Gaussian and Bernoulli distributions into a superposition of uniform noise, giving further improvement for those cases, and putting them on par with the uniform cases.

Quantum walks on random lattices: Diffusion, localization, and the absence of parametric quantum speedup

Discrete-time quantum walks, quantum generalizations of classical random walks, provide a framework for quantum information processing, quantum algorithms and quantum simulation of condensed matter systems. The key property of quantum walks, which lies at the heart of their quantum information applications, is the possibility for a parametric quantum speed-up in propagation compared to classical random walks. In this work we study propagation of quantum walks on percolation-generated two-dimensional random lattices. In large-scale simulations of topological and trivial split-step walks, we identify distinct pre-diffusive and diffusive behaviors at different time scales. Importantly, we show that even arbitrarily weak concentrations of randomly removed lattice sites give rise to a complete breakdown of the superdiffusive quantum speed-up, reducing the motion to ordinary diffusion. By increasing the randomness, quantum walks eventually stop spreading due to Anderson localization. Near the localization threshold, we find that the quantum walks become subdiffusive. The fragility of quantum speed-up implies dramatic limitations for quantum information applications of quantum walks on random geometries and graphs.

The mechanism of pyroelectricity in polar material hemimorphite

It is known that a crystal structure and symmetry determine the physical properties of materials. Lattice distortion can strongly affect the symmetry of the crystal structure. Polar materials show changes in polarization with temporal fluctuations of temperature due to the asymmetry. As a polar crystal, hemimorphite shows excellent pyroelectric properties. However, to date, there are a few studies on its intrinsic physical properties, and the mechanism of its pyroelectricity remains unclear. In this paper, single-crystal x-ray diffraction measurement was carried out to obtain the atomic positions at 100–400 K. Furthermore, the electric dipole moments of [ZnO4] and [SiO4] polyhedrons along a, b, and c axes have been calculated. The calculated pyroelectric coefficient derived from the intrinsic electric dipole moment was compared with the experimental measurement. The results indicate that the pyroelectric coefficients of hemimorphite at different temperatures mainly come from the variation of the electric dipole moment of [ZnO4] and [SiO4] polyhedrons along the c axis. The electric dipole moment changes as a function of temperature from 100 to 400 K, which is induced by the random lattice distortion. It is found that pyroelectricity is strongly correlated with the random lattice distortion. The establishment of the relationship between lattice distortion and pyroelectricity helps us to regulate the specific electrical parameters of the material, which may lead to future work in energy harvesting and further properties.

Prediction of lattice distortion and mechanical behavior of tetragonal phase (Bi0.2Na0.2Ba0.2Sr0.2Ca0.2)TiO3 high-entropy perovskite with A-site disorder from first-principles calculations

High-entropy perovskite ceramics have drawn widespread attention as their excellent physical properties. Herein, the high-entropy (Bi0.2Na0.2Ba0.2Sr0.2Ca0.2)TiO3 (BNBSCT) perovskite is selected as case study. The distorted structure and mechanical behavior as well as the effect of pressure are comprehensively studied through implementing first-principles calculations in conjunction with special quasi-random structure. The results reveal the atomic random distribution and size different cause serious lattice distortion within the BNBSCT structure, and the pseudo-intralayer distortion is significantly smaller than the interlayer distortion. Further studies of the mechanical properties indicate that the high-entropy BNBSCT perovskite is mechanical stable, especially larger elastic parameters imply the larger resistance against elastic deformation in comparison with Bi0.5Na0.5TiO3. Compared to the rule of mixture, the higher resistance to volume deformation and crack propagation of BNBSCT suggests the strengthening of high-entropy perovskite presumably due to lattice distortion in structure. Increasing pressure leads to decrease in lattice parameter and volume of structure, then increases the degree of lattice distortion, elastic constants and elastic moduli, as well the degree of lattice distortion. Moreover, the ductility and fracture toughness are also enhanced, though the elastic anisotropy is slightly larger. This work provides a new insight into the high-entropy perovskite under high pressure, which will contribute to the further research and application of high-entropy materials.

Two-dimensional diffusive epidemic process in the presence of quasiperiodic and quenched disorder

This work considers the diffusive epidemic process model coupled to the square lattice, the Penrose quasiperiodic lattice, and the Voronoi–Delaunay random lattice. The main objective is to verify if spatial disorder influences critical behavior. According to the Harris–Barghathi–Vojta criterion, quenched or quasiperiodic disorder can change the critical behavior of the system, depending on the disorder decay exponent of the lattice. We employed extensive Monte Carlo simulations of the relevant quantities. Furthermore, we estimate the critical exponent ratios. Our results suggest that the disorder does not change the critical behavior when comparing the critical exponent ratios for the three studied lattice structures. In addition, the critical exponents depend on the three possible diffusion regimes: (1) where diffusion is dominated by susceptible individuals, (2) where infected and susceptible individuals have the same diffusion constant, and (3) where diffusion is dominated by the infected individuals.

Variational quantum solutions to the Shortest Vector Problem

A fundamental computational problem is to find a shortest non-zero vector in Euclidean lattices, a problem known as the Shortest Vector Problem (SVP). This problem is believed to be hard even on quantum computers and thus plays a pivotal role in post-quantum cryptography. In this work we explore how (efficiently) Noisy Intermediate Scale Quantum (NISQ) devices may be used to solve SVP. Specifically, we map the problem to that of finding the ground state of a suitable Hamiltonian. In particular, (i) we establish new bounds for lattice enumeration, this allows us to obtain new bounds (resp. estimates) for the number of qubits required per dimension for any lattices (resp. random q-ary lattices) to solve SVP; (ii) we exclude the zero vector from the optimization space by proposing (a) a different classical optimisation loop or alternatively (b) a new mapping to the Hamiltonian. These improvements allow us to solve SVP in dimension up to 28 in a quantum emulation, significantly more than what was previously achieved, even for special cases. Finally, we extrapolate the size of NISQ devices that is required to be able to solve instances of lattices that are hard even for the best classical algorithms and find that with approximately 103 noisy qubits such instances can be tackled.

The six-vertex model on random planar maps revisited

We address the six vertex model on a random lattice, which in combinatorial terms corresponds to the enumeration of weighted 4-valent planar maps equipped with an Eulerian orientation. This problem was exactly, albeit non-rigorously solved by Ivan Kostov in 2000 using matrix integral techniques. We convert Kostov's work to a combinatorial argument involving functional equations coming from recursive decompositions of the maps, which we solve rigorously using complex analysis. We then investigate modular properties of the solution, which lead to simplifications in certain special cases. In particular, in two special cases of combinatorial interest we rederive the formulae discovered by Bousquet-Mélou and the first author.