Ginibre Point Process Research Articles

On the Characteristic Polynomial of the Eigenvalue Moduli of Random Normal Matrices

AbstractWe study the characteristic polynomial $$p_{n}(x)=\prod _{j=1}^{n}(|z_{j}|-x)$$ p n ( x ) = ∏ j = 1 n ( | z j | - x ) where the $$z_{j}$$ z j are drawn from the Mittag–Leffler ensemble, i.e. a two-dimensional determinantal point process which generalizes the Ginibre point process. We obtain precise large n asymptotics for the moment generating function $$\mathbb {E}[e^{\frac{u}{\pi } \, \text {Im\,}\ln p_{n}(r)}e^{a \, \text {Re\,}\ln p_{n}(r)}]$$ E [ e u π Im\, ln p n ( r ) e a Re\, ln p n ( r ) ] , in the case where r is in the bulk, $$u \in \mathbb {R}$$ u ∈ R and $$a \in \mathbb {N}$$ a ∈ N . This expectation involves an $$n \times n$$ n × n determinant whose weight is supported on the whole complex plane, is rotation-invariant, and has both jump- and root-type singularities along the circle centered at 0 of radius r. This “circular" root-type singularity differs from earlier works on Fisher–Hartwig singularities, and surprisingly yields a new kind of ingredient in the asymptotics, the so-called associated Hermite polynomials.

Spatial modeling and reliability analyzing of reconfigurable intelligent surfaces-assisted Fog-RAN with repulsion

Reconfigurable Intelligent Surface (RIS), fog computing, and Cell-Free (CF) network architecture are three promising technologies for application to the Ultra-Reliable Low Latency Communication (URLLC) scenario in 6G mobile communication systems. This paper considers a RIS-assisted FogRadio Access Network (Fog-RAN) architecture where a) the repulsively distributed Fog-Access Points (F-APs) communicate in a CF manner to suppress inter-cell interference, b) RISs are introduced into the CF network to avoid shadowing and enhance the system performance, and c) fog computing evolved as cloud services providers at the edge of the network and an enabler for constructing a multi-layer computing power RAN. Then, we derive and validate the integral form of the maximum F-AP offloading probability and Successful Delivery Probability (SDP) of this RIS-assisted Fog-RAN over composite Fisher-Snedecor F fading, where the spatial effects are reconsidered with the assumption that the F-APs are modelled as a Beta Ginibre Point Process (β-GPP). The numeric and simulation results indicate that for the investigated RIS-assisted Fog-RAN, the β-GPP-based deployment of F-APs can increase maximum of 8% of the SDP within the repulsion-effective range, compared with the Matern Cluster Process (MCP)-based ones. Also, deploying more RISs per F-AP offers more significant SDP improvements.

Large gap asymptotics on annuli in the random normal matrix model

We consider a two-dimensional determinantal point process arising in the random normal matrix model and which is a two-parameter generalization of the complex Ginibre point process. In this paper, we prove that the probability that no points lie on any number of annuli centered at 0 satisfies large n asymptotics of the form exp(C1n2+C2nlogn+C3n+C4n+C5logn+C6+Fn+O(n-112)),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\exp \\Bigg ( C_{1} n^{2} + C_{2} n \\log n + C_{3} n + C_{4} \\sqrt{n} + C_{5}\\log n + C_{6} + {\\mathcal {F}}_{n} + \\mathcal {O}\\Big ( n^{-\\frac{1}{12}}\\Big )\\Bigg ), \\end{aligned}$$\\end{document}where n is the number of points of the process. We determine the constants C1,…,C6\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_{1},\\ldots ,C_{6}$$\\end{document} explicitly, as well as the oscillatory term Fn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {F}}_{n}$$\\end{document} which is of order 1. We also allow one annulus to be a disk, and one annulus to be unbounded. For the complex Ginibre point process, we improve on the best known results: (i) when the hole region is a disk, only C1,…,C4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_{1},\\ldots ,C_{4}$$\\end{document} were previously known, (ii) when the hole region is an unbounded annulus, only C1,C2,C3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_{1},C_{2},C_{3}$$\\end{document} were previously known, and (iii) when the hole region is a regular annulus in the bulk, only C1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_{1}$$\\end{document} was previously known. For general values of our parameters, even C1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_{1}$$\\end{document} is new. A main discovery of this work is that Fn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {F}}_{n}$$\\end{document} is given in terms of the Jacobi theta function. As far as we know this is the first time this function appears in a large gap problem of a two-dimensional point process.

FD-JCAS Techniques for mmWave HetNets: Ginibre Point Process Modeling and Analysis

In this paper, we study the co-design of full-duplex (FD) radio with joint communication and radar sensing (JCAS) techniques in millimeter-wave (mmWave) heterogeneous networks (HetNets). Spectral co-existence of radar and communication systems causes mutual interference between the two systems, compromising both the data exchange and sensing capabilities. Focusing on the detection performance, we propose a cooperative detection technique, which exploits the sensing information from multiple base stations (BSs), aiming at enhancing the probability of successfully detecting an object. Three combining rules are considered, namely the <i>OR</i> , the <i>Majority</i> and the <i>AND</i> rule. In real-world network scenarios, the locations of the BSs are spatially correlated, exhibiting a repulsive behavior. Therefore, we model the spatial distribution of the BSs as a <inline-formula><tex-math notation="LaTeX">$\beta$</tex-math></inline-formula> -Ginibre point process ( <inline-formula><tex-math notation="LaTeX">$\beta$</tex-math></inline-formula> -GPP), which can characterize the repulsion among the BSs. By using stochastic geometry tools, analytical expressions for the detection performance of <inline-formula><tex-math notation="LaTeX">$\beta$</tex-math></inline-formula> -GPP-based FD-JCAS systems are expressed for each of the considered combining rule. Furthermore, by considering temporal interference correlation, we evaluate the probability of successfully detecting an object over two different time slots. Our results demonstrate that our proposed technique can significantly improve the detection performance when compared to the conventional non-cooperative technique.

Determinantal shot noise Cox processes

We present a new class of cluster point process models, which we call determinantal shot noise Cox processes (DSNCP), with repulsion between cluster centres. They are the special case of generalized shot noise Cox processes where the cluster centres are determinantal point processes. We establish various moment results and describe how these can be used to easily estimate unknown parameters in two particularly tractable cases, namely, when the offspring density is isotropic Gaussian and the kernel of the determinantal point process of cluster centres is Gaussian or like in a scaled Ginibre point process. Through a simulation study and the analysis of a real point pattern data set, we see that when modelling clustered point patterns, a much lower intensity of cluster centres may be needed in DSNCP models as compared to shot noise Cox processes.

Asymptotics of determinants with a rotation-invariant weight and discontinuities along circles

We study the moment generating function of the disk counting statistics of a two-dimensional determinantal point process which generalizes the complex Ginibre point process. This moment generating function involves an n×n determinant whose weight is supported on the whole complex plane, is rotation-invariant, and has discontinuities along circles centered at 0. These discontinuities can be thought of as a two-dimensional analogue of jump-type Fisher-Hartwig singularities. In this paper, we obtain large n asymptotics for this determinant, up to and including the term of order n−12. We allow for any finite number of discontinuities in the bulk, one discontinuity at the edge, and any finite number of discontinuities bounded away from the bulk. As an application, we obtain the large n asymptotics of all the cumulants of the disk counting function up to and including the term of order n−12, both in the bulk and at the edge. This improves on the best known results for the complex Ginibre point process, and for general values of our parameters these results are completely new. Our proof makes a novel use of the uniform asymptotics of the incomplete gamma function.

Linear-regression based performance approximation for millimeter-wave multicell networks with β-Ginibre deployed base stations

This paper investigates the performance of millimeter-wave (mmWave) multicell networks with repulsively deployed base stations (BSs), via the β -Ginibre point process ( β -GPP) where β adjusts the BS repulsion. In particular, this study analytically expresses the SINR distributions and validates the results via extensive simulations. Further, this paper empirically demonstrates that the effects of BS repulsion can be well modeled via linear regression (LR) and this LR-based approach can be also quite simply approximated as the convex combination (CC) of the two extreme cases of β → 0 and β = 1 . Therefore, the proposed CC approximation can significantly reduce the time for calculating the SINR distributions in β -GPP mmWave multicell networks.

Correction to: Absolute continuity and singularity of Palm measures of the Ginibre point process

Correction to: Absolute continuity and singularity of Palm measures of the Ginibre point process

Caching modeling and energy analyzing of cell-free wireless heterogeneous networks with beta Ginibre point process

Cell-free Wireless Heterogeneous Networks (HetNets) have emerged as a technological alternative for conventional cellular networks. In this paper, we study the spatially correlative caching strategy, the energy analysis, and the impact of parameter β on the total energy cost of the cell-free wireless HetNets with Access Points distributed by Beta Ginibre Point Process (β-GPP). We derive the approximate expression of Successful Delivery Probability (SDP) based on the Signal-to-Interference-plus-Noise Ratio coverage model. From both analytical and simulation results, it is shown that the proposed caching model based on β–GPP placement, which jointly takes into account path loss, fading, and interference, can closely simulate the caching performance of the cell-free HetNets in terms of SDP. By guaranteeing the outage probability constraints, the analytical expression of the uplink energy cost is also derived. Another conclusion is that with AP locations modeled by β-GPP, the power consumption is not sensitive to β, but is sensitive to the dimension of the kernel function; hence β is less restrictive, and only the truncation of the Ginibre kernel has to be appropriately modified. These findings are new compared with the existing literature where the nodes are commonly assumed to be of Poisson Point Process, Matérn Hard-Core Process, or Poisson Cluster Process deployment in cell-free systems.

Performance Analysis of Clustered Wireless-Powered Ad Hoc Networks via β-Ginibre Point Processes

We consider a wireless-powered ad hoc network with clustered Power Beacons (PBs). The wireless-powered transmitters (WP-Txs) only activate stochastically deployed PBs within the circles centered at their locations. The PBs transmit radio frequency (RF) signals, which are harvested by the WP-Txs to transmit information to their target receivers. The nodes are modeled as following <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\beta $ </tex-math></inline-formula> -Ginibre Point Processes ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\beta $ </tex-math></inline-formula> -GPP), which incorporate repulsion among PBs or active WP-Txs. Lacking analytical mathematical tools to describe the distribution of clustered PBs, we establish lower and upper bounds on the energy outage probability, as well as an approximated expression of it. We then derive the information outage probability, an approximate expression of it by approximating the reduced Palm distribution of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\beta $ </tex-math></inline-formula> -GPP as an inhomogeneous Poisson point process (IPPP), and an analytical approximate expression of it by further ignoring the small-scale fading of interfering WP-Tx links. Finally, the transmission capacity is obtained based on the overall outage probability. The simulation results show that the derived theoretical expressions accurately reflect the performance of the wireless-powered ad hoc networks and the performance gap introduced by approximations is negligible. Transmission capacity optimization based on the derived expressions is also possible and demonstrated.

Stochastic geometry and dynamics of infinitely many particle systems—random matrices and interacting Brownian motions in infinite dimensions

We explain the general theories involved in solving an infinite-dimensional stochastic differential equation (ISDE) for interacting Brownian motions in infinite dimensions related to random matrices. Typical examples are the stochastic dynamics of infinite particle systems with logarithmic interaction potentials such as the sine, Airy, Bessel, and also for the Ginibre interacting Brownian motions. The first three are infinite-dimensional stochastic dynamics in one-dimensional space related to random matrices called Gaussian ensembles. They are the stationary distributions of interacting Brownian motions and given by the limit point processes of the distributions of eigenvalues of these random matrices. The sine, Airy, and Bessel point processes and interacting Brownian motions are thought to be geometrically and dynamically universal as the limits of bulk, soft edge, and hard edge scaling. The Ginibre point process is a rotation- and translation-invariant point process on R 2 \mathbb {R}^2 , and an equilibrium state of the Ginibre interacting Brownian motions. It is the bulk limit of the distributions of eigenvalues of non-Hermitian Gaussian random matrices. When the interacting Brownian motions constitute a one-dimensional system interacting with each other through the logarithmic potential with inverse temperature β = 2 \beta = 2 , an algebraic construction is known in which the stochastic dynamics are defined by the space-time correlation function. The approach based on the stochastic analysis (called the analytic approach) can be applied to an extremely wide class. If we apply the analytic approach to this system, we see that these two constructions give the same stochastic dynamics. From the algebraic construction, despite being an infinite interacting particle system, it is possible to represent and calculate various quantities such as moments by the correlation functions. We can thus obtain quantitative information. From the analytic construction, it is possible to represent the dynamics as a solution of an ISDE. We can obtain qualitative information such as semi-martingale properties, continuity, and non-collision properties of each particle, and the strong Markov property of the infinite particle system as a whole. Ginibre interacting Brownian motions constitute a two-dimensional infinite particle system related to non-Hermitian Gaussian random matrices. It has a logarithmic interaction potential with β = 2 \beta = 2 , but no algebraic configurations are known.The present result is the only construction.

Computation Offloading Analysis in Clustered Fog Radio Access Networks With Repulsion

In this work, we leverage the clustered <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\beta$</tex-math></inline-formula> -Ginibre point process (GPP) and the queueing theory to analyse the delay performance in a large-scale repulsive fog radio access network (F-RAN) with computation offloading, which plays a key role on the delay performance. In particular, the average delay defined as the mean value of delay for executing the computing tasks and the task completion probability (TCP) defined as the probability of the delay for executing the computing task being smaller than a target delay, are derived in analytical forms and validated by Monte Carlo simulations. Based on these results, the optimal rate threshold, which affects the task offloading probability and the transmission delay, and the optimal cloud offloading probability to minimize the average delay and maximize the TCP are jointly analysed numerically. Moreover, an approximated result of the average delay is provided to reduce its computational complexity. The results show that the repulsion in the F-RAN significantly decreases the average delay and increases the TCP, but equipped with the optimal rate threshold and the cloud offloading probability, the effects of repulsion on the average delay and the TCP become trivial. Furthermore, with the optimal cloud offloading probability and rate threshold, the average delay and the TCP can be significantly reduced and improved in the hybrid processing mode compared with those in the local processing mode, respectively. The optimal rate thresholds lie in the range of 3 to 13 Mbps under our simulation environment, which can reduce a maximum of 60% of the average delay and increase a minimum of 0.17 of the TCP.

Feasibility study of 28 GHz and 38 GHz millimeter-wave technologies for fog radio access networks using multi-slope path loss model

Fog computing is considered a promising technology to reduce latency and network congestion. Meanwhile, Millimeter-wave (mmWave) communication owing to its potential for multi-gigabit of wireless channel capacity could be employed to further improve the performance of fog computing networks. In this context, we study the feasibility of using 28 GHz and 38 GHz mmWaves in fog radio access networks (F-RANs). The multi-slope path loss model is used to calculate the interference impacts because it provides a more accurate approximation of the wireless links. Simulations are carried out for uplink scenario considering the following fog node (FN) deployment models: Poisson point process (PPP), Ginibre point process (GPP), square grid, and ultra-dense network (UDN). The results depict that at low FN densities the massive accumulation of interference components severely impacts the performance. However, the performance can be improved by increasing the FN density and choosing a deployment strategy with high degree of regularity. Based on the results, we verify that it is feasible to use 28 GHz and 38 GHz mmWaves in F-RANs when the density of the interfering users is less than 150 user/km2 where capacities higher than 1 Gbps are achieved for all considered scenarios.

Continuum percolation and stochastic epidemic models on Poisson and Ginibre point processes

The most studied continuum percolation model in two dimensions is the Boolean model consisting of disks with the same radius whose centers are randomly distributed on the Poisson point process (PPP). We also consider the Boolean percolation model on the Ginibre point process (GPP) which is a typical repelling point process realizing hyperuniformity. We think that the PPP approximates a disordered configuration of individuals, while the GPP does a configuration of citizens adopting a strategy to keep social distancing in a city in order to avoid contagion. We consider the SIR models with contagious infection on supercritical percolation clusters formed on the PPP and the GPP. By numerical simulations, we studied dependence of the percolation phenomena and the infection processes on the PPP- and the GPP-underlying graphs. We show that in a subcritical regime of infection rate the PPP-based models show emergence of infection clusters on clumping of points which is formed by fluctuation of uncorrelated Poissonian statistics. On the other hand, the cumulative numbers of infected individuals in processes are suppressed in the GPP-based models.

Couplings for determinantal point processes and their reduced Palm distributions with a view to quantifying repulsiveness

AbstractFor a determinantal point process (DPP) X with a kernel K whose spectrum is strictly less than one, André Goldman has established a coupling to its reduced Palm process $X^u$ at a point u with $K(u,u)&gt;0$ so that, almost surely, $X^u$ is obtained by removing a finite number of points from X. We sharpen this result, assuming weaker conditions and establishing that $X^u$ can be obtained by removing at most one point from X, where we specify the distribution of the difference $\xi_u: = X\setminus X^u$. This is used to discuss the degree of repulsiveness in DPPs in terms of $\xi_u$, including Ginibre point processes and other specific parametric models for DPPs.

FD-JCAS Techniques for mmWave HetNets: Ginibre Point Process Modeling and Analysis

In this paper, we study the co-design of full-duplex (FD) radio with joint communication and radar sensing (JCAS) techniques in millimeter-wave (mmWave) heterogeneous networks (HetNets). Spectral co-existence of radar and communication systems causes mutual interference between the two systems, compromising both the data exchange and sensing capabilities. Focusing on the detection performance, we propose a cooperative detection technique, which exploits the sensing information from multiple base stations (BSs), aiming at enhancing the probability of successfully detecting an object. Three combining rules are considered, namely the \textit{OR}, the \textit{Majority} and the \textit{AND} rule. In real-world network scenarios, the locations of the BSs are spatially correlated, exhibiting a repulsive behavior. Therefore, we model the spatial distribution of the BSs as a $\beta$-Ginibre point process ($\beta$-GPP), which can characterize the repulsion among the BSs. By using stochastic geometry tools, analytical expressions for the detection performance of $\beta$-GPP-based FD-JCAS systems are expressed for each of the considered combining rule. Furthermore, by considering temporal interference correlation, we evaluate the probability of successfully detecting an object over two different time slots. Our results demonstrate that our proposed technique can significantly improve the detection performance when compared to the conventional non-cooperative technique.

Performance Analysis of Millimeter Wave SWIPT System Using Ginibre Point Process

This paper deals with the coverage probability analysis of mmWave simultaneous wireless information and power transfer (SWIPT) system using stochastic geometry. mmWaves are higher frequency waves results in dense deployment of base stations (BS) leading to a higher data rate. Stochastic geometry provides a mathematical model to the randomly distributed nodes in the network. The Poisson point process (PPP) is the most analytically tractable and widely used model for the location of BSs, but it fails to capture the underlying separation between the BSs of cellular networks. Ginibre point process (GPP) model is used for the repulsion between the two nodes and provides a realistic model compared to PPP. In this paper, mmWave interfering base stations are considered to be GPP for deriving the coverage probability analysis. The proposed model considers the interfering base stations for analyzing the coverage probability. The closed form expression for the coverage probability is derived and is analyzed. The theoretical results are validated through simulation results.

Optimal transport between determinantal point processes and application to fast simulation

Two optimal transport problems between determinantal point processes (DPP for short) are investigated. It is shown how to estimate the Kantorovitch–Rubinstein and Wasserstein-2 distances between distributions of DPP. These results are applied to evaluate the accuracy of a fast but approximate simulation algorithm of the Ginibre point process restricted to a circle. One can now simulate in a reasonable amount of time more than ten thousands points.

Local laws and rigidity for Coulomb gases at any temperature

We study Coulomb gases in any dimension $\mathsf{d}\geq2$ and in a broad temperature regime. We prove local laws on the energy, separation and number of points down to the microscopic scale. These yield the existence of limiting point processes after extraction, generalizing the Ginibre point process for arbitrary temperature and dimension. The local laws come together with a quantitative expansion of the free energy with a new explicit error rate in the case of a uniform background density. These estimates have explicit temperature dependence, allowing to treat regimes of very large or very small temperature, and exhibit a new minimal lengthscale for rigidity and screening, depending on the temperature. They apply as well to energy minimizers (formally zero temperature). The method is based on a bootstrap on scales and reveals the additivity of the energy modulo surface terms, via the introduction of subadditive and superadditive approximate energies.

Performance Study for Random Access-Based Wireless Mutual Broadcast Networks With Ginibre Point Processes

While the random access based wireless mutual broadcast (RA-WMB) is expected to be key to various applications including proximal data dissemination and neighbor discovery, such RA-WMB applications may have a wide range of node distributions. Therefore, this letter investigates the impact of spatial node distributions on the RA-WMB performance and design, through modeling node distributions as the β-Ginibre point process (GPP) that expresses the node repulsion via 0 <; β ≤ 1. The results demonstrate that the β-GPP node repulsion may degrade the RA-WMB performance and it rarely affects the RA-WMB operation design for improving the performance.