In the first part of this thesis I consider site and bond percolation on a Random Connection Model and prove that for a wide range of connection functions the critical site probability is strictly greater than the critical bond probability and use this fact to improve previously known non-strict inequalities to strict inequalities. In the second part I consider percolation on the even phase of a Random Sequential Adsorption model and prove that the critical intensity is finite and strictly bigger than 1. Both of these main results make use of an enhancement technique.

This dissertation investigates the influence of surface heterogeneities on colloid deposition. First, deposition of colloidal particles on a nanofiltration membrane during cross flow membrane filtration was studied under different operating pressures and solution chemistries. An atomic force microscope (AFM) was then used to observe the deposit morphology formed on the membrane. At the initial stages of fouling, more particles preferentially accumulate near the peaks than in the valleys of the rough nanofiltration membrane surface. This study demonstrates that it is difficult to isolate, correlate and assess the effects that physical (roughness) heterogeneity and chemical heterogeneity has on colloid deposition based on experiments involving surfaces where the physical and chemical heterogeneities are uncorrelated or randomly distributed. In the second phase of the study, the deposition of model colloidal particles onto patterned charge-heterogeneous surfaces was studied both experimentally and theoretically. Controlled charge heterogeneity was created experimentally employing self assembled monolayers of alkanethiols patterned onto gold substrates using a soft lithographic technique. Model colloidal particles and fluorescent nanoparticles were sequentially deposited onto the patterned substrate under no flow (quiescent) conditions, and the deposited structures and the micro-patterns were imaged in situ using a combination of phase contrast and fluorescence microscopy. This study indicates that particles tend to preferentially deposit at the edges of the chemically favourable stripes. The theoretical investigation involved the formulation of a mathematical model based on Random Sequential Adsorption (RSA). This study showed that a simple binary probability distribution assumed in the model is able to predict the experimental deposit morphology adequately, particularly the periodicity of the underlying patterns on the substrate. Furthermore, the effect of charge heterogeneity on the electrostatic double layer interaction between a particle and a charge heterogeneous planar surface was studied numerically employing a 3D finite element model. In this system, significant lateral forces at close separation distances were observed, and found to be appreciably higher when the particle is near the edge of a heterogeneous region of the substrate. From the above studies, it can be concluded that by altering/controlling the chemical heterogeneity of the substrate, it is possible to achieve significant control on the resulting deposit morphology.

No description available.

No description available.

Ce mémoire est consacré à trois modèles classiques de géométrie aléatoire : les mosaïques, les enveloppes convexes et le modèle booléen. Dans la première partie, on étudie les mosaïques poissonniennes d'hyperplans isotropes et plus particulièrement leur zéro-cellule qui est un polyèdre convexe aléatoire de l'espace euclidien. Deux cas particuliers de zéro-cellules sont la cellule typique de Poisson-Voronoi et la cellule de Crofton. On donne une formule explicite pour la loi du nombre de côtés d'une zéro-cellule en dimension deux. On s'intéresse au comportement asymptotique de cette loi et on fait le lien avec le problème de Sylvester des points en position convexe. On décrit ensuite la loi du rayon circonscrit ainsi que le comportement asymptotique du polyèdre à grand rayon inscrit au moyen de théorèmes limites. De cette manière et aussi par l'utilisation de la fréquence fondamentale, on apporte des précisions à l'énoncé de la conjecture de D. G. Kendall. La seconde partie a pour objet les enveloppes convexes de processus ponctuels de Poisson isotropes dans la boule-unité. On établit un résultat de type grandes déviations pour le nombre de sommets. On montre ensuite la convergence de la frontière de l'enveloppe après changement d'échelle et on en déduit des résultats de valeurs extrêmes, estimations de variance, théorèmes centraux limites et principes d'invariance pour certaines caractéristiques. Dans la troisième partie, on s'intéresse enfin aux modèles de recouvrement de type booléen de l'espace euclidien. Dans un premier travail, on applique une variante du modèle sans interpénétration des objets à la modélisation d'un phénomène de fissuration. On étudie ensuite la convergence de la composante connexe de l'origine d'un modèle booléen vers la cellule de Crofton en dimension deux. On s'intéresse enfin à la fonction de visibilité de cette composante connexe pour laquelle on obtient une estimée de la queue de distribution et des résultats de valeurs extrêmes.

Owing to its ability to improve both spectral and energy efficiency of wireless networks, massive multiple-input multiple-output (mMIMO) has become one of the key enablers of the fifth-generation (5G) and beyond communication systems. For successful integration of this promising physical layer technique in the upcoming cellular standards, it is essential to have a comprehensive understanding of its network-level performance. Over the last decade, stochastic geometry has been instrumental in obtaining useful system design insights of wireless networks through accurate and tractable theoretical analysis. Hence, it is only natural to consider modeling and analyzing the mMIMO systems using appropriate statistical constructs from the stochastic geometry literature and gain insights for its future implementation. With this broader objective in mind, we first focus on modeling a cellular mMIMO network that uses fractional pilot reuse to mitigate the sole performance-limiting factor of mMIMO networks, namely, pilot contamination. Leveraging constructs from the stochastic geometry literature, such as Johnson-Mehl cells, we derive analytical expressions for the uplink (UL) signal-to-interference-and-noise ratio (SINR) coverage probability and average spectral efficiency for a random user. From our system analysis, we present a partitioning rule for the number of pilot sequences to be reserved for the cell-center and cell-edge users that improves the average cell-edge user spectral efficiency while achieving similar cell-center user spectral efficiency with respect to unity pilot reuse. In addition, using the analytical approach developed for the cell-center user performance evaluation, we study the performance of a small cell system where user and base station (BS) locations are coupled. The impact of distance-dependent UL power control on the performance of an mMIMO network with unity pilot reuse is analyzed and subsequent system design guidelines are also presented. Next, we focus on the performance analysis of the cell-free mMIMO network, which is a distributed implementation of the mMIMO system that leads to the second and third contributions of this dissertation. Similar to the cellular counterpart, the cell-free systems also suffer from pilot contamination due to the reuse of pilot sequences throughout the network. Inspired by a hardcore point process known as the random sequential adsorption (RSA) process, we develop a new distributed pilot assignment algorithm that mitigates the effect of pilot contamination by ensuring a minimum distance among the co-pilot users. This pilot assignment scheme leads to the construction of a new point process, namely the multilayer RSA process. We study the statistical properties of this point process both in one and two-dimensional spaces by deriving approximate but accurate expressions for the density and pair correlation functions. Leveraging these new results, for a cell-free network with the proposed RSA-based pilot assignment scheme, we present an analytical approach that determines the minimum number of pilots required to schedule a user with probabilistic guarantees. In addition, to benchmark the performance of the RSA-based scheme, we propose two optimization-based centralized pilot allocation schemes using linear programming principles. Through extensive numerical simulations, we validate the efficacy of the distributed and scalable RSA-based pilot assignment scheme compared to the proposed centralized algorithms. Apart from pilot contamination, another impediment to the performance of a cell-free mMIMO is limited fronthaul capacity between the baseband unit and the access points (APs). In our fourth contribution, using appropriate stochastic geometry-based tools, we model and analyze the downlink of such a network for two different implementation scenarios. In the first scenario, we consider a finite network where each AP serves all the users in the network. In the second scenario, we consider an infinite network where each user is served by a few nearby APs in order to limit the load on fronthaul links. From our analyses, we observe that for the finite network, the achievable average system sum-rate is a strictly quasi-concave function of the number of users in the network, which serves as a key guideline for scheduler design for such systems. Further, for the user-centric architecture, we observe that there exists an optimal number of serving APs that maximizes the average user rate. The fifth and final contribution of this dissertation focuses on the potential improvement that is possible by the use of mMIMO in citizen broadband radio service (CBRS) spectrum sharing systems. As a first concrete step, we present comprehensive modeling and analysis of this system with omni-directional transmissions. Our model takes into account the key guidelines by the Federal Communications Commission for co-existence between licensed and unlicensed networks in the 3.5 GHz CBRS frequency band. Leveraging the properties of the Poisson hole process and Matern hardcore point process of type II, a.k.a. ghost RSA process, we analytically characterize the impact of different system parameters on various performance metrics such as medium access probability, coverage probability, and area spectral efficiency. Further, we provide useful system design guidelines for successful co-existence between these networks. Building upon this omni-directional model, we also characterize the performance benefits of using mMIMO in such a spectrum sharing network. / Doctor of Philosophy / The emergence of cloud-based video and audio streaming services, online gaming platforms, instantaneous sharing of multimedia contents (e.g., photos, videos) through social networking platforms, and virtual collaborative workspace/meetings require the cellular communication networks to provide high data-rate as well as reliable and ubiquitous connectivity. These constantly evolving requirements can be met by designing a wireless network that harmoniously exploits the symbiotic co-existence among different types of cutting-edge wireless technologies. One such technology is massive multiple-input multiple-output (mMIMO), whose core idea is to equip the cellular base stations (BSs) with a large number of antennas that can be leveraged through appropriate signal processing algorithms to simultaneously accommodate multiple users with reduced network interference. For successful deployment of mMIMO in the upcoming cellular standards, i.e., fifth-generation (5G) and beyond systems, it is necessary to characterize its performance in a large-scale wireless network taking into account the inherent spatial randomness in the BS and user locations. To achieve this goal, in this dissertation, we propose different statistical methods for the performance analysis of mMIMO networks using tools from stochastic geometry, which is a field of mathematics related to the study of random patterns of points. One of the major deployment issues of mMIMO systems is pilot contamination, which is a form of coherent network interference that degrades user performance. The main reason behind pilot contamination is the reuse of pilot sequences, which are a finite number of known signal waveforms used for channel estimation between a user and its serving BS. Further, the effect of pilot contamination is more severe for the cell-edge users, which are farther from their own BSs. An efficient scheme to mitigate the effect of pilot contamination is fractional pilot reuse (FPR). However, the efficiency of this scheme depends on the pilot partitioning rule that decides the fraction of total pilot sequences that should be used by the cell-edge users. Using appropriate statistical constructs from the stochastic geometry literature, such as Johnson-Mehl cells, we present a partitioning rule for efficient implementation of the FPR scheme in a cellular mMIMO network. Next, we focus on the performance analysis of the cell-free mMIMO network. In contrast to the cellular network, where each user is served by a single BS, in a cell-free network each user can be served by multiple access points (APs), which have less complex hardware compared to a BS. Owing to this cooperative and distributed implementation, there are no cell-edge users. Similar to the cellular counterpart, the cell-free systems also suffer from pilot contamination due to the reuse of pilot sequences throughout the network. Inspired by a hardcore point process known as the random sequential adsorption (RSA) process, we develop a new distributed pilot assignment algorithm that mitigates the effect of pilot contamination by ensuring a minimum distance among the co-pilot users. Further, we show that the performance of this distributed pilot assignment scheme is appreciable compared to different centralized pilot assignment schemes, which are algorithmically more complex and difficult to implement in a network. Moreover, this pilot assignment scheme leads to the construction of a new point process, namely the multilayer RSA process. We derive the statistical properties of this point process both in one and two-dimensional spaces. Further, in a cell-free mMIMO network, the APs are connected to a centralized baseband unit (BBU) that performs the bulk of the signal processing operations through finite capacity links, such as fiber optic cables. Apart from pilot contamination, another implementational issue associated with the cell-free mMIMO systems is the finite capacity of fronthaul links that results in user performance degradation. Using appropriate stochastic geometry-based tools, we model and analyze this network for two different implementation scenarios. In the first scenario, we consider a finite network where each AP serves all the users in the network. In the second scenario, we consider an infinite network where each user is served by a few nearby APs. As a consequence of this user-centric implementation, for each user, the BBU only needs to communicate with fewer APs thereby reducing information load on fronthaul links. From our analyses, we propose key guidelines for the deployment of both types of scenarios. The type of mMIMO systems that are discussed in this work will be operated in the sub-6 GHz frequency range of the electromagnetic spectrum. Owing to the limited availability of spectrum resources, usually, spectrum sharing is encouraged among different cellular operators in such bands. One such example is the citizen broadband radio service (CBRS) spectrum sharing systems proposed by the Federal Communications Commission (FCC). The final contribution of this dissertation focuses on the potential improvement that is possible by the use of mMIMO in the CBRS systems. As our first step, using tools from stochastic geometry, we model and analyze this system with a single antenna at the BSs. In our model, we take into account the key guidelines by the FCC for co-existence between licensed and unlicensed operators. Leveraging properties of the Poisson hole process and hardcore process, we provide useful theoretical expressions for different performance metrics such as medium access probability, coverage probability, and area spectral efficiency. These results are used to obtain system design guidelines for successful co-existence between these networks. We further highlight the potential improvement in the user performance with multiple antennas at the unlicensed BS.