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This paper uses the theory of point processes and stochastic geometry to quantify the sky visibility experienced by users located in an urban environment. The general idea is to represent the buildings of this environment as a stationary marked point process, where the points represent the building locations and the marks their heights. The point process framework is first used to characterize the distribution of the blockage angle, which limits the visibility of a typical user into the sky due to the obstruction by buildings. In the context of communications, this distribution is useful when users try to connect to the nodes of an aerial or non-terrestrial network in a Line-of-Sight way. Within this context, the point process framework can also be used to investigate the gain of connectivity obtained thanks to Reconfigurable Intelligent Surfaces. Assuming that such surfaces are installed on the top of buildings to extend the user's sky visibility, this point process approach allows one to quantify the gain in visibility and hence the gain in connectivity obtained by the typical user. The distributional properties of visibility-related metrics are cross-validated by comparison to simulation results., Comment: 13 pages, 13 figures, 2 tables, submitted to IEEE

Consider a stationary sequence $X=(X_n)$ of integer-valued random variables with mean $m \in [-\infty, \infty]$. Let $S=(S_n)$ be the stochastic process with increments $X$ and such that $S_0=0$. For each time $i$, draw an edge from $(i,S_i)$ to $(j,S_j)$, where $j>i$ is the smallest integer such that $S_j \geq S_i$, if such a $j$ exists. This defines the record graph of $X$. It is shown that if $X$ is ergodic, then its record graph exhibits the following phase transitions when $m$ ranges from $-\infty$ to $\infty$. For $m<0$, the record graph has infinitely many connected components which are all finite trees. At $m=0$, it is either a one-ended tree or a two-ended tree. For $m>0$, it is a two-ended tree. The distribution of the component of $0$ in the record graph is analyzed when $X$ is an i.i.d. sequence of random variables whose common distribution is supported on $\{-1,0,1,\ldots\}$, making $S$ a skip-free to the left random walk. For this random walk, if $m<0$, then the component of $0$ is a unimodular typically re-rooted Galton-Watson Tree. If $m=0$, then the record graph rooted at $0$ is a one-ended unimodular random tree, specifically, it is a unimodular Eternal Galton-Watson Tree. If $m>0$, then the record graph rooted at $0$ is a unimodularised bi-variate Eternal Kesten Tree. A unimodular random directed tree is said to be record representable if it is the component of $0$ in the record graph of some stationary sequence. It is shown that every infinite unimodular ordered directed tree with a unique succession line is record representable. In particular, every one-ended unimodular ordered directed tree has a unique succession line and is thus record representable., Comment: Added a paragraph in Page 38 in the proof of Theorem 53, added Example 59 (Unimodular EFT with two succession lines) page 42

The reconfigurable intelligent surface (RIS) technology allows one to engineer spatial diversity in complex cellular networks. This paper provides a framework for the system-level performance assessment of RIS-assisted networks and in particular downlink coverage probability and ergodic rate. To account for the inherent randomness in the spatial deployments of base stations (BSs) and RISs, we model the placements of the RISs as point processes (PPs) conditioned on the associated BSs, which are modeled by a Poisson point process (PPP). These RIS PPs can be adapted based on the deployment strategy. We focus on modeling the RISs as a Mat\'ern cluster process (MCP), where each RIS cluster is a finite PPP with support a disc centered on the association BS. We assume that the system uses the orthogonal frequency division multiplexing (OFDM) technique to exploit the multipath diversity provided by RISs. The coverage probability and the ergodic rate can be evaluated when RISs operate as batched powerless beamformers. The resulting analytical expressions provide a general methodology to evaluate the impact of key RIS-related parameters, such as the batch size and the density of RISs, on system-level performance. To demonstrate the framework's broad applicability, we also analyze a RIS placement variant where RISs are deployed around coverage holes. Numerical evaluations of the analytical expressions and Monte-Carlo simulations jointly validate the proposed analytical approach and provide valuable insights into the design of future RIS-assisted cellular networks.

We introduce a stochastic geometry dynamics inspired by opinion dynamics, which captures the essence of modern asymmetric social networks, with leaders and followers. Opinions are represented by points in the Euclidean space and the leader of an agent is the agent with the closest opinion. In this dynamics, each follower updates its opinion by halving the distance to its leader. We show that this simple dynamics and its iterates exhibit several interesting purely geometric phenomena pertaining to the evolution of leadership and of opinion clusters, which are reminiscent of those observed in social network. We also show that when the initial opinions are randomly distributed as a stationary Poisson point process, the probability of occurrence of each of these phenomena can be represented by an integral geometry formula involving semi-agebraic domains. We finally analyze numerically the limiting behavior of this follower dynamics. In the Poisson case, the agents fall into two categories, that of ultimate followers, which never stop updating their opinions, and that of ultimate leaders, which adopt a fixed opinion in a finite time. All our findings are complemented by spatial discrete event simulations.

With the growing interest in satellite networks, satellite-terrestrial integrated networks (STINs) have gained significant attention because of their potential benefits. However, due to the lack of a tractable network model for the STIN architecture, analytical studies allowing one to investigate the performance of such networks are not yet available. In this work, we propose a unified network model that jointly captures satellite and terrestrial networks into one analytical framework. Our key idea is based on Poisson point processes distributed on concentric spheres, assigning a random height to each point as a mark. This allows one to consider each point as a source of desired signal or a source of interference while ensuring visibility to the typical user. Thanks to this model, we derive the probability of coverage of STINs as a function of major system parameters, chiefly path-loss exponent, satellites and terrestrial base stations' height distributions and density, transmit power and biasing factors. Leveraging the analysis, we concretely explore two benefits that STINs provide: i) coverage extension in remote rural areas and ii) data offloading in dense urban areas., Comment: submitted to IEEE journal

This work develops a novel approach toward performance guarantees for all links in arbitrarily large wireless networks. It introduces a spatial network calculus, consisting of spatial regulation properties for stationary point processes and the first steps of a calculus for this regulation, which can be seen as an extension to space of the classical network calculus. Specifically, two classes of regulations are defined: one includes ball regulation and shot-noise regulation, which are shown to be equivalent and upper constraint interference; the other one includes void regulation, which lower constraints the signal power. These regulations are defined both in the strong and weak sense: the former requires the regulations to hold everywhere in space, whereas the latter only requires the regulations to hold as observed by a jointly stationary point process. Using this approach, we derive performance guarantees in device-to-device, ad hoc, and cellular networks under proper regulations. We give universal bounds on the SINR for all links, which gives link service guarantees based on information-theoretic achievability. They are combined with classical network calculus to provide end-to-end latency guarantees for all packets in wireless queuing networks. Such guarantees do not exist in networks that are not spatially regulated, e.g., Poisson networks., Comment: Submitted to IEEE Transactions on Wireless Communications

To model existing or future low Earth orbit (LEO) satellite networks leveraging multiple constellations, we propose a simple analytical approach to represent the clustering of satellites on orbits. More precisely, we develop a variable-altitude Poisson orbit process that effectively captures the geometric fact that satellites are always positioned on orbits, and these orbits may vary in altitude. Conditionally on the orbit process, satellites situated on these orbits are modeled as linear Poisson point processes, thereby forming a Cox point process. For this model, we derive useful statistics, including the distribution of the distance from the typical user to its nearest visible satellite, the outage probability, the Laplace functional of the proposed Cox satellite point process, and the Laplace transform of the interference power from the Cox-distributed satellites under general fading. The derived statistics enable the evaluation of the performance of such LEO satellite communication systems as functions of network parameters., Comment: accepted to IEEE Trans. Veh. Technol

This work develops an analytical framework for downlink low Earth orbit (LEO) or medium Earth orbit (MEO) satellite communications, leveraging tools from stochastic geometry. We propose a tractable approach to the analysis of such satellite communication systems, accounting for the fact that satellites are located on circular orbits. We accurately incorporate this geometric property of LEO or MEO satellite constellations by developing a Cox point process model that jointly produces orbits and satellites on these orbits. Our work contrasts with previous modeling studies that presumed satellite locations to be entirely random, thereby overlooking the fundamental fact that satellites are jointly positioned on orbits. Employing this Cox model, we analyze the network performance experienced by users located on Earth. Specifically, we evaluate the no-satellite probability of the proposed network and the Laplace transform of the interference created by such a network. Using it, we compute its SIR (signal-to-interference) distribution, namely its coverage probability. By presenting fundamental network performance as functions of key parameters, this model allows one to assess the statistical properties of downlink LEO or MEO satellite communications and can thus be used as a system-level design tool to operate and optimize forthcoming complex LEO or MEO satellite networks., Comment: Submitted to IEEE Journal

Consider a migration process based on a closed network of N stations with K_N customers. Each station is a ./M/\infty queue with service (migration) rate mu. Upon departure, a customer is routed at random to another station. In addition to migration, these customers are subject to an SIS (Susceptible, Infected, Susceptible) dynamics: customers are either I for infected, or S for susceptible. They can swap their state either from I to S or from S to I only in stations. At any station, each S customer becomes I with rate alpha Y if there are Y infected customers in the station, and each I customer recovers and becomes S with rate beta. We let N tend to infinity and assume that lim_{N\to infty} K_N/N= eta>0. The main problem is about the set of parameters for which there exists a stationary regime where the epidemic survives in the thermodynamic limit. We establish several structural properties of the system, which allow us to give the phase transition diagram of this thermodynamic limit w.r.t. eta. The analysis of the SIS model reduces to that of a wave-type PDE for which we found no explicit solution. This SIS model is one among several companion stochastic processes with migration and contagion. Two of them are discussed as they provide some bounds and approximations to SIS. These two variants are the DOCS (Departure On Change of State) and the AIR (Averaged Infection Rate), which both admit closed-form solutions. The AIR system is a mean-field model where the infection mechanism is based on the empirical average of the number of infected customers in all stations. The latter admits a product-form solution. DOCS features accelerated migration in that each change of SIS state implies an immediate departure. It leads to another wave-type PDE that admits a closed-form solution.

This paper studies a multiclass spatial birth-and-death (SBD) processes on a compact region of the Euclidean plane modeling wireless interactions. In this model, users arrive at a constant rate and leave at a rate function of the interference created by other users in the network. The novelty of this work lies in the addition of service differentiation, inspired by bandwidth partitioning present in 5G networks: users are allocated a fixed number of frequency bands and only interfere with transmissions on these bands. The first result of the paper is the determination of the critical user arrival rate below which the system is stochastically stable, and above which it is unstable. The analysis requires symmetry assumptions which are defined in the paper. The proof for this result uses stochastic monotonicity and fluid limit models. The monotonicity allows one to bound the dynamics from above and below by two adequate discrete-state Markov jump processes, for which we obtain stability and instability results using fluid limits. This leads to a closed form expression for the critical arrival rate. The second contribution consists in two heuristics to estimate the steady-state densities of all classes of users in the network: the first one relies on a Poisson approximation of the steady-state processes. The second one uses a cavity approximation leveraging second-order moment measures, which leads to more accurate estimates of the steady-state user densities. The Poisson heuristic also gives a good estimate for the critical arrival rate., Comment: Need some reformulations

In vehicle-to-all (V2X) communications, roadside units (RSUs) play an essential role in connecting various network devices. In some cases, users may not be well-served by RSUs due to congestion, attenuation, or interference. In these cases, vehicular relays associated with RSUs can be used to serve those users. This paper uses stochastic geometry to model and analyze a spatially correlated heterogeneous vehicular network where both RSUs and vehicular relays serve network users such as pedestrians or other vehicles. We present an analytical model where the spatial correlation between roads, RSUs, relays, and users is systematically modeled via Cox point processes. Assuming users are associated with either RSUs or relays, we derive the association probability and the coverage probability of the typical user. Then, we derive the user throughput by considering interactions of links unique to the proposed network. This paper gives practical insights into designing spatially correlated vehicular networks assisted by vehicle relays. For instance, we express the network performance such as the user association, SIR coverage probability, and the network throughput as the functions of network key geometric variables. In practice, this helps one to optimize the network so as to achieve ultra reliability or maximum user throughput of a spatially correlated vehicular networks by varying key aspects such as the relay density or the bandwidth for relays., Comment: submitted to IEEE Journal

The Doeblin Graph of a countable state space Markov chain describes the joint pathwise evolutions of the Markov dynamics starting from all possible initial conditions, with two paths coalescing when they reach the same point of the state space at the same time. Its Bridge Doeblin subgraph only contains the paths starting from a tagged point of the state space at all possible times. In the irreducible, aperiodic, and positive recurrent case, the following results are known: the Bridge Doeblin Graph is an infinite tree that is unimodularizable. Moreover, it contains a single bi-infinite path, which allows one to build a perfect sample of the stationary state of the Markov chain. The present paper is focused on the null recurrent case. It is shown that when assuming irreducibility and aperiodicity again, the Bridge Doeblin Graph is either an infinite tree or a forest made of a countable collection of infinite trees. In the first case, the infinite tree in question has a single end, is not unimodularizable in general, but is always locally unimodular. These key properties are used to study the stationary regime of several measure-valued random dynamics on this Bridge Doeblin Tree. The most important ones are the taboo random dynamics, which admits as steady state a random measure with mean measure equal to the invariant measure of the Markov chain, and the potential random dynamics which is a random extension of the classical potential measure, with a mean measure equal to infinity at every point of the state space., Comment: 31 Pages, 8 figures

We develop an analytical framework to examine the line-of-sight (LOS) coverage area in vehicular networks with roadside units (RSU) and vehicle relays. In practical deployment scenarios, RSUs and vehicle relays are spatially correlated and we characterize this by employing Cox point processes to model the locations of RSUs and vehicle relays simultaneously. Leveraging the random blockage model, we model the LOS coverage area as Boolean models on these Cox point processes. The LOS coverage area is then evaluated by its area fraction. We show that relays can increase the area fraction of LOS coverage by nearly 50\% even when RSUs and relays are spatially correlated. By presenting a stochastic geometry model for a vehicular network with RSUs and relays and then by providing a tool to capture its LOS coverage, our work assesses the viability of vehicle relays for modern vehicular networks exploiting LOS coverage., Comment: accepted IEEE Trans. Veh. Technol

Inspired by a new feature in 5G NR called bandwidth part (BWP), this paper presents a bandwidth allocation (BA) model that allows one to adapt the bandwidth allocated to users depending on their data rate needs. Specifically, in adaptive BA, a wide bandwidth is divided into chunks of smaller bandwidths and the number of bandwidth chunks allocated to a user depends on its needs or type. Although BWP in 5G NR mandates allocation of a set of contiguous bandwidth chunks, our BA model also allows other assumptions on chunk allocation such as the allocation of any set of bandwidth chunks, as in, e.g., LTE resource allocation, where chunks are selected uniformly at random. The BA model studied here is probabilistic in that the user locations are assumed to form a realization of a Poisson point process and each user decides independently to be of a certain type with some probability. This model allows one to quantify spectrum sharing and service differentiation in this context, namely to predict what performance a user gets depending on its type as well as the overall performance. This is based on exact representations of key performance metrics for each user type, namely its success probability, the meta distribution of its signal-to-interference ratio, and its Shannon throughput. We show that, surprisingly, the higher traffic variability stemming from adaptive BA is beneficial: when comparing two networks using adaptive BA and having the same mean signal and the same mean interference powers, the network with higher traffic variability performs better for all these performance metrics. With respect to Shannon throughput, we observe that our BA model is roughly egalitarian per Hertz and leads to a linear service differentiation in aggregated throughput value., Comment: IEEE INFOCOM 2020

This paper presents a novel spatially consistent approach for modeling line-of-sight (LOS) paths in vehicular networks. We use stochastic geometry to model transmitters, obstacles, and receivers located in three parallel lines, respectively. Their geometric interactions are leveraged to characterize the existence of LOS paths. Specifically, the proposed approach focuses on the role of obstacles in blocking one or more LOS paths, which has been overlooked in most statistical models for blockage. Under the proposed framework, we derive the probability that a typical vehicle is in LOS with respect to transmitters with received signal-to-noise ratios greater than a threshold. The proposed framework and LOS coverage analysis are instrumental to the analysis of LOS-critical applications such as positioning or mmWave communications in vehicular networks., Comment: Submitted to IEEE Journal

Beam management is central in the operation of beamformed wireless cellular systems such as 5G New Radio (NR) networks. Focusing the energy radiated to mobile terminals (MTs) by increasing the number of beams per cell increases signal power and decreases interference, and has hence the potential to bring major improvements on area spectral efficiency (ASE). This paper proposes a first system-level stochastic geometry model encompassing major aspects of the beam management problem: frequencies, antenna configurations, and propagation; physical layer, wireless links, and coding; network geometry, interference, and resource sharing; sensing, signaling, and mobility management. This model leads to a simple analytical expression for the effective rate that the typical user gets in this context. This in turn allows one to find the number of beams per cell and per MT that maximizes the effective ASE by offering the best tradeoff between beamforming gains and beam management operational overheads and costs, for a wide variety of 5G network scenarios including millimeter wave (mmWave) and sub-6 GHz. As part of the system-level analysis, we define and analyze several underlying new and fundamental performance metrics that are of independent interest. The numerical results discuss the effects of different systemic tradeoffs and performance optimizations of mmWave and sub-6 GHz 5G deployments., Comment: This is an extended version of the paper that was accepted at IEEE GLOBECOM 2020. Comments are welcome. arXiv admin note: text overlap with arXiv:2006.05027

Consider 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 one is the set of points that belong to some one-dimensional facet of the Voronoi tessellation and are 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 is about the Palm distribution of the angle with which the points of this second point process see the two nuclei associated with the facet. The note gives answers to these two questions and briefly discusses their practical motivations. It also discusses natural extensions to dimension three.

This paper considers the time evolution of a queue that is embedded in a Poisson point process of moving wireless interferers. The queue is driven by an external arrival process and is subject to a time-varying service process that is a function of the SINR that it sees. Static configurations of interferers result in an infinite queue workload with positive probability. In contrast, a generic stability condition is established for the queue in the case where interferers possess any non-zero mobility that results in displacements that are both independent across interferers and oblivious to interferer positions. The proof leverages the mixing property of the Poisson point process. The effect of an increase in mobility on queueing metrics is also studied. Convex ordering tools are used to establish that faster moving interferers result in a queue workload that is smaller for the increasing-convex stochastic order. As a corollary, mean workload and mean delay decrease as network mobility increases. This stochastic ordering as a function of mobility is explained by establishing positive correlations between SINR level-crossing events at different time points, and by determining the autocorrelation function for interference and observing that it decreases with increasing mobility. System behaviour is empirically analyzed using discrete-event simulation and the performance of various mobility models is evaluated using heavy-traffic approximations., Comment: Preliminary version appeared in WiOPT 2020. New version with revisions

This paper is focused on SIS (Susceptible-Infected-Susceptible) epidemic dynamics (also known as the contact process) on populations modelled by homogeneous Poisson point processes of the Euclidean plane, where the infection rate of a susceptible individual is proportional to the number of infected individuals in a disc around it. The main focus of the paper is a model where points are also subject to some random motion. Conservation equations for moment measures are leveraged to analyze the stationary regime of the point processes of infected and susceptible individuals. A heuristic factorization of the third moment measure is then proposed to obtain simple polynomial equations allowing one to derive closed form approximations for the fraction of infected individuals in the steady state. These polynomial equations also lead to a phase diagram which tentatively delineates the regions of the space of parameters (population density, infection radius, infection and recovery rate, and motion rate) where the epidemic survives and those where there is extinction. A key take-away from this phase diagram is that the extinction of the epidemic is not always aided by a decrease in the motion rate. These results are substantiated by simulations on large two dimensional tori. These simulations show that the polynomial equations accurately predict the fraction of infected individuals when the epidemic survives. The simulations also show that the proposed phase diagram accurately predicts the parameter regions where the mean survival time of the epidemic increases (resp. decreases) with motion rate., Comment: post-revision, JMB

We study an N-player game where a pure action of each player is to select a non-negative function on a Polish space supporting a finite diffuse measure, subject to a finite constraint on the integral of the function. This function is used to define the intensity of a Poisson point process on the Polish space. The processes are independent over the players, and the value to a player is the measure of the union of its open Voronoi cells in the superposition point process. Under randomized strategies, the process of points of a player is thus a Cox process, and the nature of competition between the players is akin to that in Hotelling competition games. We characterize when such a game admits Nash equilibria and prove that when a Nash equilibrium exists, it is unique and comprised of pure strategies that are proportional in the same proportions as the total intensities. We give examples of such games where Nash equilibria do not exist. A better understanding of the criterion for the existence of Nash equilibria remains an intriguing open problem., Comment: Preliminary versions presented at the Workshop on Information Theory and its Applications, ITA-2019, San Diego, in February 2019 and the Symposium on Advances in Communications Networks, IISC, Bengaluru, in July 2020

Beam management is central in the operation of dense 5G cellular networks. Focusing the energy radiated to mobile terminals (MTs) by increasing the number of beams per cell increases signal power and decreases interference, and has hence the potential to bring major improvements on area spectral efficiency (ASE). This benefit, however, comes with unavoidable overheads that increase with the number of beams and the MT speed. This paper proposes a first system-level stochastic geometry model encompassing major aspects of the beam management problem: frequencies, antennas, and propagation; physical layer, wireless links, and coding; network geometry, interference, and resource sharing; sensing, signaling, and mobility management. This model leads to a simple analytical expression for the effective ASE that the typical user gets in this context. This in turn allows one to find, for a wide variety of 5G network scenarios including millimeter wave (mmWave) and sub-6 GHz, the number of beams per cell that offers the best global trade-off between these benefits and costs. We finally provide numerical results that discuss the effects of different systemic trade-offs and performances of mmWave and sub-6 GHz 5G deployments., Comment: This is a work in progress. Your comments are welcome. 9 pages, 6 figures

We study how dense multi-antenna millimeter wave (mmWave) cellular network performance scales in terms of the base station (BS) spatial density $\lambda$, by studying the signal-to-interference-plus-noise ratio (SINR) and the area spectral efficiency (ASE). If the number of antennas at each BS scales at least linearly with $\lambda$, which increases the number of possible beam configurations and their main-lobe gain, and decreases their side-lobe gain, we prove that the SINR approaches a finite random variable that is independent of $\lambda$ and the ASE scales at least linearly with $\lambda$. In contrast, if the number of antennas scales sub-linearly with $\lambda$, then the SINR decays to zero and the ASE saturates to a constant. Thus, by moving to higher carrier frequencies with successively smaller antennas, and exploiting the correspondingly increased directionality, cellular operators can in principle avoid the densification plateau (or collapse) in cellular networks and instead continue to harvest linear sum throughput gains through BS densification.

Network dynamics with point-process-based interactions are of paramount modeling interest. Unfortunately, most relevant dynamics involve complex graphs of interactions for which an exact computational treatment is impossible. To circumvent this difficulty, the replica-mean-field approach focuses on randomly interacting replicas of the networks of interest. In the limit of an infinite number of replicas, these networks become analytically tractable under the so-called "Poisson Hypothesis". However, in most applications, this hypothesis is only conjectured. Here, we establish the Poisson Hypothesis for a general class of discrete-time, point-process-based dynamics, that we propose to call fragmentation-interaction-aggregation processes, and which are introduced in the present paper. These processes feature a network of nodes, each endowed with a state governing their random activation. Each activation triggers the fragmentation of the activated node state and the transmission of interaction signals to downstream nodes. In turn, the signals received by nodes are aggregated to their state. Our main contribution is a proof of the Poisson Hypothesis for the replica-mean-field version of any network in this class. The proof is obtained by establishing the propagation of asymptotic independence for state variables in the limit of an infinite number of replicas. Discrete time Galves-L\"{o}cherbach neural networks are used as a basic instance and illustration of our analysis.

Replica-mean-field models have been proposed to decipher the activity of neural networks via a multiply-and-conquer approach. In this approach, one considers limit networks made of infinitely many replicas with the same basic neural structure as that of the network of interest, but exchanging spikes in a randomized manner. The key point is that these replica-mean-field networks are tractable versions that retain important features of the finite structure of interest. To date, the replica framework has been discussed for first-order models, whereby elementary replica constituents are single neurons with independent Poisson inputs. Here, we extend this replica framework to allow elementary replica constituents to be composite objects, namely, pairs of neurons. As they include pairwise interactions, these pair-replica models exhibit non-trivial dependencies in their stationary dynamics, which cannot be captured by first-order replica models. Our contributions are two-fold: $(i)$ We analytically characterize the stationary dynamics of a pair of intensity-based neurons with independent Poisson input. This analysis involves the reduction of a boundary-value problem related to a two-dimensional transport equation to a system of Fredholm integral equations---a result of independent interest. $(ii)$ We analyze the set of consistency equations determining the full network dynamics of certain replica limits. These limits are those for which replica constituents, be they single neurons or pairs of neurons, form a partition of the network of interest. Both analyses are numerically validated by computing input/output transfer functions for neuronal pairs and by computing the correlation structure of certain pair-dominated network dynamics., Comment: 40 pages, 6 figures

This paper concerns the performance of vehicle-to-everything (V2X) communications. More precisely, we analyze the broadcast of safety-related V2X communications in cellular networks where base stations and vehicles are assumed to share the same spectrum and vehicles broadcast their safety messages to neighboring users. We model the locations of vehicles as a Poisson line Cox point process and the locations of users as a planar Poisson point process. We assume that users are associated with their closest base stations when there is no vehicle within a certain distance $ \rho $. On the other hand, users located within a distance $ \rho $ from vehicles are associated with the vehicles to receive their safety messages. We quantify the properties of this vehicle-prioritized association using the stochastic geometry framework. We derive the fractions of users that receive safety messages from vehicles. Then, we obtain the expression for the signal-to-interference ratio of the typical user evaluated on each association type. To address the impact of vehicular broadcast on the cellular network, the paper also derives the effective rate offered to the typical user in this setting., Comment: Submitted to IEEE Trans. Wireless Commun

We study the scaling laws of the signal-to-interference-plus-noise ratio (SINR) and area spectral efficiency (ASE) in multi-antenna cellular networks, where the number of antennas scales with the base station (BS) spatial density $\lambda$. We start with the MISO case having $N_t(\lambda)$ transmit antennas and a single receive antenna and prove that the average SINR scales as $\frac{N_t(\lambda)}{\lambda}$ and the average ASE scales as $\lambda\log\left(1+\frac{N_t(\lambda)}{\lambda}\right)$. For the MIMO case with single-stream eigenbeamforming and $N_r(\lambda) \leq N_t(\lambda)$ receive antennas, we prove that the scaling laws of the conditional SINR and ASE are exactly the same as the MISO case, i.e. not dependent on $N_r(\lambda)$. We also show that coordinated beamforming amongst $K\leq N_t(\lambda)$ neighboring BSs does not improve the scaling laws regardless of $K$. From a system design perspective, our results suggest that deploying multi-antenna BSs can help maintain the per-user throughput and the linear increase in the ASE with BS density, while the number of antennas at the user equipment and the use of BS cooperation do not matter much., Comment: 28 pages. arXiv admin note: substantial text overlap with arXiv:2001.05083

We study the scaling laws of the signal-to-interference-plus-noise ratio (SINR) and the area spectral efficiency (ASE) in multi-antenna cellular networks, where the number of antennas scales with the base station (BS) spatial density $\lambda$, under the assumption of independent and identically distributed (i.i.d.) channels. We start with the MISO case with $N_t(\lambda)$ transmit antennas and a single receive antenna and prove that the average SINR scales as $\frac{N_t(\lambda)}{\lambda}$ and the average ASE scales as $\lambda\log\left(1+\frac{N_t(\lambda)}{\lambda}\right)$. For the MIMO case with single-stream eigenbeamforming and $N_r(\lambda) \leq N_t(\lambda)$ receive antennas, we prove that the scaling laws of the conditional SINR and ASE are agnostic to $N_r(\lambda)$ and scale exactly the same as the MISO case. Hence, deploying multi-antenna BSs can help maintain non-zero per-user throughput and a corresponding linear increase in the ASE in dense cellular networks.

Background: Haplotypes, the ordered lists of single nucleotide variations that distinguish chromosomal sequences from their homologous pairs, may reveal an individual's susceptibility to hereditary and complex diseases and affect how our bodies respond to therapeutic drugs. Reconstructing haplotypes of an individual from short sequencing reads is an NP-hard problem that becomes even more challenging in the case of polyploids. While increasing lengths of sequencing reads and insert sizes {\color{black} helps improve accuracy of reconstruction}, it also exacerbates computational complexity of the haplotype assembly task. This has motivated the pursuit of algorithmic frameworks capable of accurate yet efficient assembly of haplotypes from high-throughput sequencing data. Results: We propose a novel graphical representation of sequencing reads and pose the haplotype assembly problem as an instance of community detection on a spatial random graph. To this end, we construct a graph where each read is a node with an unknown community label associating the read with the haplotype it samples. Haplotype reconstruction can then be thought of as a two-step procedure: first, one recovers the community labels on the nodes (i.e., the reads), and then uses the estimated labels to assemble the haplotypes. Based on this observation, we propose ComHapDet - a novel assembly algorithm for diploid and ployploid haplotypes which allows both bialleleic and multi-allelic variants. Conclusions: Performance of the proposed algorithm is benchmarked on simulated as well as experimental data obtained by sequencing Chromosome $5$ of tetraploid biallelic \emph{Solanum-Tuberosum} (Potato). The results demonstrate the efficacy of the proposed method and that it compares favorably with the existing techniques., Comment: To Appear in BMC Genomics. Extended Abstract in ACM CNB-MAC 2019

We characterize the stability, metastability, and the stationary regime of traffic dynamics in a single-cell uplink wireless system. The traffic is represented in terms of spatial birth-death processes, in which users arrive as a Poisson point process in time and space, each with a file to transmit to the base station. The service rate of each user is based on its signal to interference plus noise ratio, where the interference is from other active users in the cell. Once the file is fully transmitted, the user leaves the cell. We derive the necessary and sufficient condition for network stability, which is independent of the specific bounded path loss function. A novel observation is that for a certain range of arrival rates, the network appears stable for a possibly long time, and then suddenly exhibits instability. This property, which is known in statistical physics but rarely observed in wireless communication, is called metastability. Finally, we propose two heuristic characterizations based on mean-field interpretation, of the network steady-state regime when it exists. The first-order approximation is very simple to compute, but loose in some regimes, whereas the second-order approximation is more sophisticated but tight for the whole range of arrival rates.

This paper explores an emerging wireless Internet-of-things (IoT) architecture based on unmanned aerial vehicles (UAVs). We consider a network where a fleet of UAVs at a fixed altitude flies on planned trajectories and IoT devices on the ground are scheduled to transmit their data to the UAVs when the latter are nearby. In such a system, the UAVs' motion triggers the uplink transmissions of the IoT devices. As a result, network performance is determined by the geometric and dynamic characteristics of the system. We propose a joint stationary model for UAVs and IoT devices and then evaluate the interference, the coverage probability, and the data rate of the typical UAV. To assess the harvesting capability of the proposed architecture, we derive a formula for the amount of data uploaded from each IoT device to a UAV. We also establish a linear relationship between the UAV coverage and the harvesting capability of the network, which provides insights into the design of the proposed harvesting scheme. In addition, we use our analytical results to numerically show that there exists a trade-off between the uploaded data and the size of the IoT scheduling window. Specifically, for a given UAV and IoT geometry, there exists an optimal scheduling window that maximizes the harvesting capability of the proposed network., Comment: IEEE Trans. Wireless Commun. (short version in Proc. IEEE ISIT)

Neural computations emerge from myriads of neuronal interactions occurring in intricate spiking networks. Due to the inherent complexity of neural models, relating the spiking activity of a network to its structure requires simplifying assumptions, such as considering models in the thermodynamic mean-field limit. In the thermodynamic mean-field limit, an infinite number of neurons interact via vanishingly small interactions, thereby erasing the finite size of interactions. To better capture the finite-size effects of interactions, we propose to analyze the activity of neural networks in the replica-mean-field limit. Replica-mean-field models are made of infinitely many replicas which interact according to the same basic structure as that of the finite network of interest. Here, we analytically characterize the stationary dynamics of an intensity-based neural network with spiking reset and heterogeneous excitatory synapses in the replica-mean-field limit. Specifically, we functionally characterize the stationary dynamics of these limit networks via ordinary differential equations derived from the Poisson Hypothesis of queuing theory. We then reduce this functional characterization to a system of self-consistency equations specifying the stationary neuronal firing rates. Of general applicability, our approach combines rate-conservation principles from point-process theory and analytical considerations from generating-function methods. We validate our approach by demonstrating numerically that replica-mean-field models better capture the dynamics of neural networks with large, sparse connections than their thermodynamic counterparts. Finally, we explain that improved performance by analyzing the neuronal rate-transfer functions, which saturate due to finite-size effects in the replica-mean-field limit., Comment: 40 pages, 3 figures

This paper proposes a framework to analyze an emerging wireless architecture where vehicles collect data from devices. Using stochastic geometry, the devices are modeled by a planar Poisson point process. Independently, roads and vehicles are modeled by a Poisson line process and a Cox point process, respectively. For any given time, a vehicle is assumed to communicate with a roadside device in a disk of radius $ \nu $ centered at the vehicle, which is referred to as the coverage disk. We study the proposed network by analyzing its short-term and long-term behaviors based on its space and time performance metrics, respectively. As short-term analysis, we explicitly derive the signal-to-interference ratio distribution of the typical vehicle and the area spectral efficiency of the proposed network. As long-term analysis, we derive the area fraction of the coverage disks and then compute the latency of the network by deriving the distribution of the minimum waiting time of a typical device to be covered by a disk. Leveraging these properties, we analyze various trade-off relationships and optimize the network utility. We further investigate these trade-offs using comparison with existing cellular networks., Comment: Submitted to IEEE Trans. Wireless Commun

This paper is centered on the random graph generated by a Doeblin-type coupling of discrete time processes on a countable state space whereby when two paths meet, they merge. This random graph is studied through a novel subgraph, called a bridge graph, generated by paths started in a fixed state at any time. The bridge graph is made into a unimodular network by marking it and selecting a root in a specified fashion. The unimodularity of this network is leveraged to discern global properties of the larger Doeblin graph. Bi-recurrence, i.e., recurrence both forwards and backwards in time, is introduced and shown to be a key property in uniquely distinguishing paths in the Doeblin graph, and also a decisive property for Markov chains indexed by $\mathbb{Z}$. Properties related to simulating the bridge graph are also studied., Comment: 44 pages, 4 figures

A stationary stochastic geometric model is proposed for analyzing the data compression method used in one-bit compressed sensing. The data set is an unconstrained stationary set, for instance all of $\mathbb{R}^n$ or a stationary Poisson point process in $\mathbb{R}^n$. It is compressed using a stationary and isotropic Poisson hyperplane tessellation, assumed independent of the data. That is, each data point is compressed using one bit with respect to each hyperplane, which is the side of the hyperplane it lies on. This model allows one to determine how the intensity of the hyperplanes must scale with the dimension $n$ to ensure sufficient separation of different data by the hyperplanes as well as sufficient proximity of the data compressed together. The results have direct implications in compressive sensing and in source coding., Comment: 29 pages

The notions of unimodular Minkowski and Hausdorff dimensions are defined in [arXiv:1807.02980] for unimodular random discrete metric spaces. The present paper is focused on the connections between these notions and the polynomial growth rate of the underlying space. It is shown that bounding the dimension is closely related to finding suitable equivariant weight functions (i.e., measures) on the underlying discrete space. The main results are unimodular versions of the mass distribution principle, Billingsley's lemma and Frostman's lemma, which allow one to derive upper bounds on the unimodular Hausdorff dimension from the growth rate of suitable equivariant weight functions. These results allow one to compute or bound both types of unimodular dimensions in a large set of examples in the theory of point processes, unimodular random graphs, and self-similarity. Further results of independent interest are also presented, like a version of the max-flow min-cut theorem for unimodular one-ended trees and a weak form of pointwise ergodic theorems for all unimodular discrete spaces., Comment: 39 pages. This paper is now merged with the latest version of arXiv:1807.02980. A previous version of this submission was titled `On the Dimension of Unimodular Discrete Spaces, Part II: Relations with Growth Rate'

This work introduces two new notions of dimension, namely the unimodular Minkowski and Hausdorff dimensions, which are inspired from the classical analogous notions. These dimensions are defined for unimodular discrete spaces, introduced in this work, which provide a common generalization to stationary point processes under their Palm version and unimodular random rooted graphs. The use of unimodularity in the definitions of dimension is novel. Also, a toolbox of results is presented for the analysis of these dimensions. In particular, analogues of Billingsley's lemma and Frostman's lemma are presented. These last lemmas are instrumental in deriving upper bounds on dimensions, whereas lower bounds are obtained from specific coverings. The notions of unimodular Hausdorff size, which is a discrete analogue of the Hausdorff measure, and unimodular dimension function are also introduced. This toolbox allows one to connect the unimodular dimensions to other notions such as volume growth rate, discrete dimension and scaling limits. It is also used to analyze the dimensions of a set of examples pertaining to point processes, branching processes, random graphs, random walks, and self-similar discrete random spaces. Further results of independent interest are also presented, like a version of the max-flow min-cut theorem for unimodular one-ended trees and a weak form of pointwise ergodic theorems for all unimodular discrete spaces., Comment: 89 pages, 1 figure. This version of the paper is a merging of the previous version with arXiv:1808.02551. Earlier versions of this paper were titled `On the Dimension of Unimodular Discrete Spaces, Part I: Definitions and Basic Properties'

Based on a simple object, an i.i.d. sequence of positive integer-valued random variables, $\{a_n\}_{n\in \mathbb{Z}}$, we introduce and study two random structures and their connections. First, a population dynamics, in which each individual is born at time $n$ and dies at time $n+a_n$. This dynamics is that of a D/GI/$\infty$ queue, with arrivals at integer times and service times given by $\{a_n\}_{n\in \mathbb{Z}}$. Second, the directed random graph $T^f$ on $\mathbb{Z}$ generated by the random map $f(n)=n+a_n$. Only assuming $\mathbb{E}[a_0]<\infty$ and $\mathbb{P}[a_0=1]>0$, we show that, in steady state, the population dynamics is regenerative, with one individual alive at each regenerative epochs. We identify a unimodular structure in this dynamics. More precisely, $T^f$ is a unimodular directed tree, in which $f(n)$ is the parent of $n$. This tree has a unique bi-infinite path. Moreover, $T^f$ splits the integers into two categories: ephemeral integers, with a finite number of descendants of all degrees, and successful integers, with an infinite number. Each regenerative epoch is a successful individual such that all integers less than it are its descendants of some order. Ephemeral, successful, and regenerative integers form stationary and mixing point processes on $\mathbb{Z}$.

This paper analyzes statistical properties of the Poisson line Cox point process useful in the modeling of vehicular networks. The point process is created by a two-stage construction: a Poisson line process to model road infrastructure and independent Poisson point processes, conditionally on the Poisson lines, to model vehicles on the roads. We derive basic properties of the point process, including the general quadratic position of the points, the nearest distance distribution, the Laplace functional, the densities of facets of the Cox-Voronoi tessellation, and the asymptotic behavior of the typical Voronoi cell under vehicular densification. These properties are closely linked to features that are important in vehicular networks., Comment: accepted to IEEE Transactions on Vehicular Technology

This paper studies the asymptotic properties of average area spectral efficiency (ASE) of a downlink cellular network in the limit of very dense base station (BS) and user densities. This asymptotic analysis relies on three assumptions: (1) interference is treated as noise; (2) the BS locations are drawn from a Poisson point process; (3) the path loss function is bounded above satisfying mild regularity conditions. We consider three possible definitions of the average ASE, all of which give units of bits per second per unit bandwidth per unit area. When there is no constraint on the minimum operational signal-to-interference-plus-noise ratio (SINR) and instantaneous full channel state information (CSI) is available at the transmitter, the average ASE is proven to saturate to a constant, which we derive in a closed form. For the other two ASE definitions, wherein either a minimum SINR is enforced or CSI is not available, the average ASE is instead shown to collapse to zero at high BS density. We provide several familiar case studies for the class of considered path loss models, and demonstrate that our results cover most previous models and results on ultradense networks as special cases., Comment: Revised version

We propose and analyze a new shadowing field model meant to capture spatial correlations. The interference field associated with this new model is compared to that of the widely used independent shadowing model. Independent shadowing over links is adopted because of the resulting closed forms for performance metrics, and in spite of the well-known fact that the shadowing fields of networks are spatially correlated. The main purpose of this paper is to challenge this independent shadowing approximation. For this, we analyze the interference measured at the origin in networks where 1) nodes which are in the same cell of some random shadowing tessellation share the same shadow, or 2) nodes which share a common mother point in some cluster process share the same shadow. By leveraging stochastic comparison techniques, we give the order relation of the three main user performance metrics, namely coverage probability, Shannon throughput and local delay, under both the correlated and the independent shadowing assumptions. We show that the evaluation of the considered metrics under the independent approximation is systematically pessimistic compared to the correlated shadowing model. The improvement in each metric when adopting the correlated shadow model is quantified and shown to be quite significant., Comment: A longer version of an accepted paper in IEEE infocom 2018, 11 pages

This paper analyzes an emerging architecture of cellular network utilizing both planar base stations uniformly distributed in Euclidean plane and base stations located on roads. An example of this architecture is that where, in addition to conventional planar cellular base stations and users, vehicles also play the role of both base stations and users. A Poisson line process is used to model the road network and, conditionally on the lines, linear Poisson point processes are used to model the vehicles on the roads. The conventional planar base stations and users are modeled by independent planar Poisson point processes. The joint stationarity of the elements in this model allows one to use Palm calculus to investigate statistical properties of such a network. Specifically, this paper discusses two different Palm distributions, with respect to the user point processes depending on its type: planar or vehicular. We derive the distance to the nearest base station, the association of the typical users, and the coverage probability of the typical user in terms of integral formulas. Furthermore, this paper provides a comprehensive characterization of the performance of all possible cellular transmissions in this setting, namely vehicle-to-vehicle (V2V), vehicle-to-infrastructure (V2I), infrastructure-to-vehicle (I2V), and infrastructure-to-infrastructure (I2I) communications., Comment: Submitted to IEEE Transactions on Communications

This paper introduces a class of non-linear and continuous-time opinion dynamics model with additive noise and state dependent interaction rates between agents. The model features interaction rates which are proportional to a negative power of opinion distances. We establish a non-local partial differential equation for the distribution of opinion distances and use Mellin transforms to provide an explicit formula for the stationary solution of the latter, when it exists. Our approach leads to new qualitative and quantitative results on this type of dynamics. To the best of our knowledge these Mellin transform results are the first quantitative results on the equilibria of opinion dynamics with distance-dependent interaction rates. The closed form expressions for this class of dynamics are obtained for the two agent case. However the results can be used in mean-field models featuring several agents whose interaction rates depend on the empirical average of their opinions. The technique also applies to linear dynamics, namely with a constant interaction rate, on an interaction graph.

Consider a countably infinite collection of interacting queues, with a queue located at each point of the $d$-dimensional integer grid, having independent Poisson arrivals, but dependent service rates. The service discipline is of the processor sharing type,with the service rate in each queue slowed down, when the neighboring queues have a larger workload. The interactions are translation invariant in space and is neither of the Jackson Networks type, nor of the mean-field type. Coupling and percolation techniques are first used to show that this dynamics has well defined trajectories. Coupling from the past techniques are then proposed to build its minimal stationary regime. The rate conservation principle of Palm calculus is then used to identify the stability condition of this system, where the notion of stability is appropriately defined for an infinite dimensional process. We show that the identified condition is also necessary in certain special cases and conjecture it to be true in all cases. Remarkably, the rate conservation principle also provides a closed form expression for the mean queue size. When the stability condition holds, this minimal solution is the unique translation invariant stationary regime. In addition, there exists a range of small initial conditions for which the dynamics is attracted to the minimal regime. Nevertheless, there exists another range of larger though finite initial conditions for which the dynamics diverges, even though stability criterion holds., Comment: Minor Spell Changes

Cell search is the process for a user to detect its neighboring base stations (BSs) and make a cell selection decision. Due to the importance of beamforming gain in millimeter wave (mmWave) and massive MIMO cellular networks, the directional cell search delay performance is investigated. A cellular network with fixed BS and user locations is considered, so that strong temporal correlations exist for the SINR experienced at each BS and user. For Poisson cellular networks with Rayleigh fading channels, a closed-form expression for the spatially averaged mean cell search delay of all users is derived. This mean cell search delay for a noise-limited network (e.g., mmWave network) is proved to be infinite whenever the non-line-of-sight (NLOS) path loss exponent is larger than 2. For interference-limited networks, a phase transition for the mean cell search delay is shown to exist in terms of the number of BS antennas/beams $M$: the mean cell search delay is infinite when $M$ is smaller than a threshold and finite otherwise. Beam-sweeping is also demonstrated to be effective in decreasing the cell search delay, especially for the cell edge users., Comment: Submitted to IEEE Transactions on Communications

We study the problem of community detection (CD) on Euclidean random geometric graphs where each vertex has two latent variables: a binary community label and a $\mathbb{R}^d$ valued location label which forms the support of a Poisson point process of intensity $\lambda$. A random graph is then drawn with edge probabilities dependent on both the community and location labels. In contrast to the stochastic block model (SBM) that has no location labels, the resulting random graph contains many more short loops due to the geometric embedding. We consider the recovery of the community labels, partial and exact, using the random graph and the location labels. We establish phase transitions for both sparse and logarithmic degree regimes, and provide bounds on the location of the thresholds, conjectured to be tight in the case of exact recovery. We also show that the threshold of the distinguishability problem, i.e., the testing between our model and the null model without community labels exhibits no phase-transition and in particular, does not match the weak recovery threshold (in contrast to the SBM)., Comment: To appear, in Information and Inference : A journal of the IMA, 2020

A key function of cloud infrastructure is to store and deliver diverse files, e.g., scientific datasets, social network information, videos, etc. In such systems, for the purpose of fast and reliable delivery, files are divided into chunks, replicated or erasure-coded, and disseminated across servers. It is neither known in general how delays scale with the size of a request nor how delays compare under different policies for coding, data dissemination, and delivery. Motivated by these questions, we develop and explore a set of evolution equations as a unified model which captures the above features. These equations allow for both efficient simulation and mathematical analysis of several delivery policies under general statistical assumptions. In particular, we quantify in what sense a workload aware delivery policy performs better than a workload agnostic policy. Under a dynamic or stochastic setting, the sample path comparison of these policies does not hold in general. The comparison is shown to hold under the weaker increasing convex stochastic ordering, still stronger than the comparison of averages. This result further allows us to obtain insightful computable performance bounds. For example, we show that in a system where files are divided into chunks of equal size, replicated or erasure-coded, and disseminated across servers at random, the job delays increase sub-logarithmically in the request size for small and medium-sized files but linearly for large files., Comment: 13 pages, 4 figures

Goldman [7] proved that the distribution of a stationary determinantal point process (DPP) $\Phi$ can be coupled with its reduced Palm version $\Phi^{0,!}$ such that there exists a point process $\eta$ where $\Phi = \Phi^{0,!} \cup \eta$ in distribution and $\Phi^{0,!} \cap \eta = \emptyset$. The points of $\eta$ characterize the repulsive nature of a typical point of $\Phi$. In this paper, the first moment measure of $\eta$ is used to study the repulsive behavior of DPPs in high dimensions. It is shown that many families of DPPs have the property that the total number of points in $\eta$ converges in probability to zero as the space dimension $n$ goes to infinity. It is also proved that for some DPPs there exists an $R^*$ such that the decay of the first moment measure of $\eta$ is slowest in a small annulus around the sphere of radius $\sqrt{n}R^*$. This $R^*$ can be interpreted as the asymptotic reach of repulsion of the DPP. Examples of classes of DPP models exhibiting this behavior are presented and an application to high dimensional Boolean models is given., Comment: 35 pages

Distance-based attenuation is a critical aspect of wireless communications. As opposed to the ubiquitous power-law path loss model, this paper proposes a stretched exponential path loss model that is suitable for short-range communication. In this model, the signal power attenuates over a distance $r$ as $e^{-\alpha r^{\beta}}$, where $\alpha,\beta$ are tunable parameters. Using experimental propagation measurements, we show that the proposed model is accurate for short to moderate distances in the range $r \in (5,300)$ meters and so is a suitable model for dense and ultradense networks. We integrate this path loss model into a downlink cellular network with base stations modeled by a Poisson point process, and derive expressions for the coverage probability, potential throughput, and area spectral efficiency. Although the most general result for coverage probability has a double integral, several special cases are given where the coverage probability has a compact or even closed form. We then show that the potential throughput is maximized for a particular BS density and then collapses to zero for high densities, assuming a fixed SINR threshold. We next prove that the area spectral efficiency, which assumes an adaptive SINR threshold, is non-decreasing with the BS density and converges to a constant for high densities., Comment: To be published in IEEE TWC

We propose an iterated version of the Gilbert model, which results in a sequence of random mosaics of the plane. We prove that under appropriate scaling, this sequence of mosaics converges to that obtained by a classical Poisson line process with explicit cylindrical measure. Our model arises from considerations on tropical plane curves, which are zeros of random tropical polynomials in two variables. In particular, the iterated Gilbert model convergence allows one to derive a scaling limit for Poisson tropical plane curves. Our work raises a number of open questions at the intersection of stochastic and tropical geometry., Comment: 36 pages, 11 figures, with appendix