Poisson Voronoi Research Articles

Poisson–Voronoi percolation in the hyperbolic plane with small intensities

Poisson–Voronoi percolation in the hyperbolic plane with small intensities

Effects of random ferroelectric and dielectric phase distributions on junctionless ferroelectric field effect transistors

In this study, we comprehensively investigated the effects of random ferroelectric (FE) and dielectric (DE) phase distributions on junctionless ferroelectric field-effect transistors (JL-FeFETs). The Poisson–Voronoi tessellation (PVT) algorithm, which corresponds to the physical growth mechanism, was used to obtain grain nucleation in the ferroelectric layer. The simulation results demonstrated that as the probability of FE phase decreased from 80% to 40%, the standard deviation of the memory window (σMW) increased from 62.4 to 99.5 mV, and the possibility of forming a blocking current path from the source to the drain increased, which degraded the memory window (MW). The simulation results indicated that decreasing the gate length and width increased device variations. Furthermore, σMW decreased from 84.5 to 58.9 mV as the grain size decreased from 5 to 3 nm.

Many-sided Poisson–Voronoi cells with only Gabriel neighbors

Let pnG be the probability for a planar Poisson–Voronoi cell to be n-sided and have only Gabriel neighbors. Using an exact coordinate transformation followed by scaling arguments and a mean-field type calculation, we obtain the asymptotic expansion of log⁡pnG in the limit of large n. We determine several statistical properties of a many-sided cell obeying this ‘Gabriel condition.’ In particular, the cell perimeter, when parametrized as a function τ(θ) of the polar angle θ, behaves as a Brownian bridge on the interval 0⩽θ⩽2π . We point out similarities and differences with related problems in random geometry.

Effective electrical conductivity of random resistor networks generated using a Poisson–Voronoi tessellation

We studied the effective electrical conductivity of dense random resistor networks (RRNs) produced using a Voronoi tessellation when its seeds are generated by means of a homogeneous Poisson point process in the two-dimensional Euclidean space. Such RRNs are isotropic and in average homogeneous; however, local fluctuations of the number of edges per unit area are inevitable. These RRNs may mimic, e.g., crack-template-based transparent conductive films. The RRNs were treated within a mean-field approach. We found an analytical dependency of the effective electrical conductivity on the number of conductive edges (resistors) per unit area, nE. The effective electrical conductivity is proportional to nE when nE≫1.

Properties of a Random Bipartite Geometric Associator Graph Inspired by Vehicular Networks.

We consider a point process (PP) generated by superimposing an independent Poisson point process (PPP) on each line of a 2D Poisson line process (PLP). Termed PLP-PPP, this PP is suitable for modeling networks formed on an irregular collection of lines, such as vehicles on a network of roads and sensors deployed along trails in a forest. Inspired by vehicular networks in which vehicles connect with their nearest wireless base stations (BSs), we consider a random bipartite associator graph in which each point of the PLP-PPP is associated with the nearest point of an independent PPP through an edge. This graph is equivalent to the partitioning of PLP-PPP by a Poisson Voronoi tessellation (PVT) formed by an independent PPP. We first characterize the exact distribution of the number of points of PLP-PPP falling inside the ball centered at an arbitrary location in R2 as well as the typical point of PLP-PPP. Using these distributions, we derive cumulative distribution functions (CDFs) and probability density functions (PDFs) of kth contact distance (CD) and the nearest neighbor distance (NND) of PLP-PPP. As intermediate results, we present the empirical distribution of the perimeter and approximate distribution of the length of the typical chord of the zero-cell of this PVT. Using these results, we present two close approximations of the distribution of node degree of the random bipartite associator graph. In a vehicular network setting, this result characterizes the number of vehicles connected to each BS, which models its load. Since each BS has to distribute its limited resources across all the vehicles connected to it, a good statistical understanding of load is important for an efficient system design. Several applications of these new results to different wireless network settings are also discussed.

Fundamentals of Vehicular Communication Networks With Vehicle Platoons

Vehicular platooning is a promising way to facilitate efficient movement of vehicles with a shared route. Despite its relevance, the interplay of platooning and the communication performance in the resulting vehicular network (VN) is largely unexplored. Inspired by this, we develop a comprehensive approach to statistical modeling and system-level analysis of VNs with platooned traffic. Modeling the network of roads using the by-now well-accepted Poisson line process (PLP), we place vehicles on each road according to an independent Matérn cluster process (MCP) that jointly captures randomness in the locations of platoons on the roads and vehicles within each platoon. The resulting <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">triply-stochastic</i> point process is a PLP-driven-Cox process, which we term the PLP-MCP. We first present this new point process’s distribution and derive several fundamental properties essential for the resulting VN’s analysis. Assuming that the cellular base-stations (BSs) are distributed as a Poisson point process (PPP), we derive the distribution of the loads served by the typical BS and the BS associated with the typical user. In deriving the latter, we also present a new approach to deriving the length distribution of a tagged chord in a Poisson Voronoi tessellation. Using the derived results, we present the rate coverage of the typical user while considering partial loading of the BSs. We also provide a comparative analysis of VNs with and without platooning of traffic.

A modeling study of transport through composite membrane with support pore location distributing randomly based on 2D Voronoi tessellations

Composite membrane consisting of selective layer and support is the mainstream membrane configuration. Support has drawn increasing attention in recent years. In this work, transport through composite membrane with support pore location distributing randomly was studied based on 2D Voronoi tessellations and the corresponding membrane performance calculation method was developed. For the calculation, firstly, irregular Voronoi regions are transformed into regular polygons with fixed number of edges meanwhile maintaining the region area. Then, the membrane performance can be obtained by combining the computation results of regular support surface pore spatial distribution and random support surface pore spatial distribution formulas. Poisson Voronoi diagrams, one of the most representative Voronoi diagrams, were mainly used to describe the support surface pore spatial distribution. The results indicate the importance of the number of support surface pores, which determines the average area of Voronoi regions. Support surface pores should be as many as possible. Another important conclusion is that a certain randomness of support surface pore structure (size and location) may not exert an influence on the composite membrane performance. That is, when calculating the composite membrane performance, the support surface pore spatial distribution and surface pore size distribution can be considered as being regular.

Upper bounds on the one-arm exponent for dependent percolation models

We prove upper bounds on the one-arm exponent $$\eta _1$$ for a class of dependent percolation models which generalise Bernoulli percolation; while our main interest is level set percolation of Gaussian fields, the arguments apply to other models in the Bernoulli percolation universality class, including Poisson–Voronoi and Poisson–Boolean percolation. More precisely, in dimension $$d=2$$ we prove that $$\eta _1 \le 1/3$$ for continuous Gaussian fields with rapid correlation decay (e.g. the Bargmann–Fock field), and in $$d \ge 3$$ we prove $$\eta _1 \le d/3$$ for finite-range fields, both discrete and continuous, and $$\eta _1 \le d-2$$ for fields with rapid correlation decay. Although these results are classical for Bernoulli percolation (indeed they are best-known in general), existing proofs do not extend to dependent percolation models, and we develop a new approach based on exploration and relative entropy arguments. The proof also makes use of a new Russo-type inequality for Gaussian fields, which we apply to prove the sharpness of the phase transition and the mean-field bound for finite-range fields.

Fairness driven performance analysis for 3-D Poisson Voronoi cell

The requirement of excellent anywhere–anytime data transmission service in future wireless network encourages us to reconsider the fairness among different types of users. To this end, the three-dimensional Poisson point process (PPP) model-based closed-form expressions of coverage probability for both cell-edge user (CEU) and cell inner user (CIU) under multi-user multiple-input multiple-output (MIMO) are derived. It is found that the spectral efficiency of CEU is about 30% of that of CIU under single user scenario. Moreover, when the number of users equals to that of base station (BS) antennas, the difference of coverage probability between CIU and CEU decreases with the number of BS antennas for large signal-to-interference ratio threshold. In addition, for the fixed number of users case, an inverted U-shaped relationship (thereby resulting in a worst case) between the fairness among CEU and CIU, and the number of BS antennas is detected. The impact of massive MIMO on the fairness under the metric of spectral efficiency is also discussed.

Individual Roles of Network and Sheet Solids in Architectured Porous Materials

To understand the different roles played by sheet solids and network solids in complex porous biomaterials/native tissues, we designed a new kind of nature-inspired structure comprising these two solids by using triply periodic minimal surfaces and Voronoi diagrams using the CAD method and compared them with the previously reported Poisson-Voronoi (PV) solids that only comprise network solids. Here, we show that the sheet solids contribute greater stiffness and solid/void interface than the network solids, and our TPMS-Voronoi solids can improve/tune both the elastic moduli and specific surfaces even at fixed solid volume fraction and can be stiffer than the Poisson-Voronoi solids. This can directly guide the porous materials design for use in tissue engineering and aerospace.

Criteria for Poisson process convergence with applications to inhomogeneous Poisson–Voronoi tessellations

This article employs the relation between probabilities of two consecutive values of a Poisson random variable to derive conditions for the weak convergence of point processes to a Poisson process. As applications, we consider the starting points of k-runs in a sequence of Bernoulli random variables, point processes constructed using inradii and circumscribed radii of inhomogeneous Poisson–Voronoi tessellations and large nearest neighbor distances in a Boolean model of disks.

Criticality and Covered Area Fraction in Confetti and Voronoi Percolation

Using the randomized algorithm method developed by Duminil-Copin et al. (Probab Theory Relat Fields 173(1–2):479–90, 2019), we exhibit sharp phase transition for the confetti percolation model. This provides an alternate proof, than that of Ahlberg et al. (Probab Theory Relat Fields 172(1–2):525–581, 2018), for the critical parameter for percolation in this model to be 1/2 when the radius of the underlying shapes for the distinct colours arise from the same distribution. In addition, we study the covered area fraction for this model, which is akin to the covered volume fraction in continuum percolation. Modulo a certain ‘transitivity condition’, this study allows us to calculate exact critical parameter for percolation when the underlying shapes for different colours may be of different sizes. Similar results are also obtained for the Poisson Voronoi percolation model when different coloured points have different growth speeds.

Palm theory, random measures and Stein couplings

We establish a general Berry–Esseen type bound which gives optimal bounds in many situations under suitable moment assumptions. By combining the general bound with Palm theory, we deduce a new error bound for assessing the accuracy of normal approximation to statistics arising from random measures, including stochastic geometry. We illustrate the use of the bound in four examples: completely random measures, excursion random measure of a locally dependent random process, and the total edge length of Ginibre–Voronoi tessellations and of Poisson–Voronoi tessellations. Moreover, we apply the general bound to Stein couplings and discuss the special cases of local dependence and additive functionals in occupancy problems.

On point processes defined by angular conditions on Delaunay neighbors in the Poisson–Voronoi Tessellation

AbstractConsider a homogeneous Poisson point process of the Euclidean plane and its Voronoi tessellation. The present note discusses the properties of two stationary point processes associated with the latter and depending on a parameter $\theta$ . The first is the set of points that belong to some one-dimensional facet of the Voronoi tessellation and such that the angle with which they see the two nuclei defining the facet is $\theta$ . The main question of interest on this first point process is its intensity. The second point process is that of the intersections of the said tessellation with a straight line having a random orientation. Its intensity is well known. The intersection points almost surely belong to one-dimensional facets. The main question here concerns the Palm distribution of the angle with which the points of this second point process see the two nuclei associated with the facet. We will give answers to these two questions and briefly discuss their practical motivations. We also discuss natural extensions to three dimensions.

Large planar Poisson–Voronoi cells containing a given convex body

Large planar Poisson–Voronoi cells containing a given convex body

Variability Analysis in a 3-D Multigranular Ferroelectric Capacitor

A simulation-based study of variability of remnant polarization $\left (P_r \right)$ in a multi-granular 3-D ultra-thin ferroelectric (FE) capacitor is presented in this paper. The Poisson Voronoi Tessellation Diagram (PVD) is used for the nucleation of grains in the FE region, which corresponds to the physical growth mechanism. The PVD algorithm implemented in MATLAB is coupled with TCAD simulations, to trace the ferroelectric hysteresis loop. It is found that the grains which have linear profile of $P_r$ show larger variability in the FE hysteresis loop, compared to the grains, which follow the Gaussian distribution of $P_r$. Additionally, the impact of dielectric content in the FE grains is analyzed. It is seen that the dielectric grains cause very large amount of variability in the FE hysteresis loop. An increase in the dielectric grains also leads to a loss in the retentivity of the hysteresis loop.

3D Network Modeling for THz-Enabled Ultra-Fast Dense Networks: A 6G Perspective

Terahertz (THz) communications are envisioned as a critical technology for 6G and enable ultra-fast dense networks with tiny cells. Implementing THz-enabled ultra-fast cells requires new techniques due to its 3D nature. Stochastic geometry is a tool that is widely used for network deployment, which provides tractability and feasibility. In this article, we provide a summary of 3D network modeling for the THz system. We discuss the new 6G ultra-cell scenario along with a potential 3D network architecture. Several recent developments in network modeling applying a stochastic geometry-based clustered process are also presented. We also discuss several potential network modeling techniques such as Poisson Voronoi (PV) tessellation, Thomas cluster process (TCP), and Matérn cluster process (MCP). Subsequently, a 3D network model for THz-band ultra-cell is demonstrated. Finally, we discuss major KPIs for 3D stochastic geometry and conclude with research challenges and open issues for 3D THz architectures.

Continuum line-of-sight percolation on Poisson–Voronoi tessellations

AbstractIn this work, we study a new model for continuum line-of-sight percolation in a random environment driven by the Poisson–Voronoi tessellation in the d-dimensional Euclidean space. The edges (one-dimensional facets, or simply 1-facets) of this tessellation are the support of a Cox point process, while the vertices (zero-dimensional facets or simply 0-facets) are the support of a Bernoulli point process. Taking the superposition Z of these two processes, two points of Z are linked by an edge if and only if they are sufficiently close and located on the same edge (1-facet) of the supporting tessellation. We study the percolation of the random graph arising from this construction and prove that a 0–1 law, a subcritical phase, and a supercritical phase exist under general assumptions. Our proofs are based on a coarse-graining argument with some notion of stabilization and asymptotic essential connectedness to investigate continuum percolation for Cox point processes. We also give numerical estimates of the critical parameters of the model in the planar case, where our model is intended to represent telecommunications networks in a random environment with obstructive conditions for signal propagation.

Underlay Radar-Massive MIMO Spectrum Sharing: Modeling Fundamentals and Performance Analysis

In this work, we study underlay radar-massive MIMO cellular coexistence in LoS/near-LoS channels, where both systems have 3D beamforming capabilities. Using mathematical tools from stochastic geometry, we derive an upper bound on the average interference power at the radar due to the 3D massive MIMO cellular downlink under the worst-case `cell-edge beamforming' conditions. To overcome the technical challenges imposed by asymmetric and arbitrarily large cells, we devise a novel construction in which each Poisson Voronoi (PV) cell is bounded by its circumcircle to bound the effect of the random cell shapes on average interference. Since this model is intractable for further analysis due to the correlation between adjacent PV cells' shapes and sizes, we propose a tractable nominal interference model, where we model each PV cell as a circular disk with an area equal to the average area of the typical cell. We quantify the gap in the average interference power between these two models and show that the upper bound is tight for realistic deployment parameters. We also compare them with a more practical but intractable MU-MIMO scheduling model to show that our worst-case interference models show the same trends and do not deviate significantly from realistic scheduler models. Under the nominal interference model, we characterize the interference distribution using the dominant interferer approximation by deriving the equi-interference contour expression when the typical receiver uses 3D beamforming. Finally, we use tractable expressions for the interference distribution to characterize radar's spatial probability of false alarm/detection in a quasi-static target tracking scenario. Our results reveal useful trends in the average interference as a function of the deployment parameters (BS density, exclusion zone radius, antenna height, transmit power of each BS, etc.).

Joint Spatial-Propagation Modeling of Cellular Networks Based on the Directional Radii of Poisson Voronoi Cells

In coverage-oriented networks, base stations (BSs) are deployed in a way such that users at the cell boundaries achieve sufficient signal strength. The shape and size of cells vary from BS to BS, since the large-scale signal propagation conditions differ in different geographical regions. This work proposes and studies a joint spatial-propagation (JSP) model, which considers the correlation between cell radii and the large-scale signal propagation (captured by shadowing). We first introduce the notion of the directional radius of Voronoi cells, which has applications in cellular networks and beyond. The directional radius of a cell is defined as the distance from the nucleus to the cell boundary at an angle relative to the direction of a uniformly random location in the cell. We study the distribution of the radii in two types of cells in the Poisson Voronoi tessellations: the zero-cell, which contains the origin, and the typical cell. The results are applied to analyze the JSP model. We show that, even though the Poisson point process (PPP) is often considered as a pessimistic spatial model for BS locations, the JSP model with the PPP achieves coverage performance close to the most optimistic one-the standard triangular lattice model. Further, we show that the network performance depends critically on the variance of the large-scale path loss along the cell boundary.