This dissertation contributes to a methodology and a better understanding that can be used to study the effects of strategies during a pandemic, especially in multi-community networks. The dissertation is structured as the following: In the first chapter, we introduce the concept of networks and its properties, and node and link percolation, which is an important process embedded in networks. Then we discuss different epidemic models, among which the SIR model is representative of many infectious diseases, and can also be mapped into a link percolation problem. We bring up two quantities that are most important in evaluating the effectiveness of epidemic strategies, one is the total fraction of individuals ever been infected by the final steady state of the SIR model, the other is the peak fraction of infected throughout the process, the second of which has seldom been studied before. There have been many researches on epidemic models within isolated networks, but recently people start getting more interested in network of networks, due to its better representation of real world systems. So we study those two quantities and their dependence on the fraction of bridge nodes in multi-community networks, in the second and third chapters: In the second chapter, we look at the final steady state of the SIR (Susceptible-Infected-Recovered) model, which can be mapped as one cluster in a link percolation problem. Using the scaling relations for the cluster size distributions around the critical point within isolated networks, we find multiple regimes in a network with two communities so that the total fraction of individuals ever been infected asymptotically follows different power laws with the fraction of bridge nodes within each regime. We also find crossovers between neighbor regimes so that the power law exponent changes from one regime to the other. It is interesting to note that the power-law relations get steeper in regimes with smaller transmissibilities, so those epidemic strategies that reduce connections between communities are more effective in those regimes. In the third chapter, we look at the peak fraction of infected of the SIR model, which also shows power law relations with the fraction of bridge nodes in different regimes, as well as crossovers between regimes. We also find that the power-law relation for the peak fraction of infected with the fraction of bridge nodes is steeper than the one for the total fraction of individuals ever been infected in the same regime, which indicates that the peak fraction of infected is more sensitive to strategies that reduce connections between communities. This explains why strategies to flatten the curve are usually taken when there are limited medical resources.

In the last three decades, the electronic industry has registered a tremendous progress. The continuous and aggressive downsizing of the transistor feature sizes (CMOS scaling) has been the main driver of the astonishing growth and advancement of microelectronic industry. Currently, the CMOS scaling is almost reaching its limits. The gate oxide is now only a few atomic layers thick, and this extremely thin oxide causes a huge leakage current through the oxide. Therefore, a further reduction of the gate oxide thickness is extremely difficult and new materials with higher dielectric constant are being explored. However, the phenomena of oxide breakdown and reliability are still serious issues in these thin oxides. Oxide breakdown exhibits a soft breakdown behavior at low voltages, and this is posing as one of the most crucial reliability issues for scaling of the ultra-thin oxides. In addition, the stress-induced leakage current (SILC) due to oxide has emerged as a scaling problem for the non-volatile memory technologies. In this dissertation, a percolation modeling approach is introduced to study and understand the dramatic changes in the conductivity of a disordered medium. Two different simulation methods of percolative conduction, the site and bond percolation, are studied here. These are used in simulating the post-breakdown conduction inside the oxide. Adopting a Monte-Carlo method, oxide breakdown is modeled using a 2-D percolation theory. The breakdown statistics and post-breakdown characteristics of the oxide are computed using this model. In this work, the effects of different physical parameters, such as dimension and the applied stress are studied. The simulation results show that a thinning of oxide layer and increasing the oxide area result in softening of breakdown. It is observed that the breakdown statistics appear to follow Weibull characteristics. As revealed by simulations, the Weibull slope changes linearly with oxide thickness, while not having a significant change when the area is varied and when the amount of the applied stress is varied. It is shown that the simulation results are well correlated with the experimental data reported in the literature. In this thesis, studying the conduction through the oxide using percolation model, it was discovered that a critical or a quasi-critical phenomenon occurs depending on the oxide dimensions. The criticality of the phase-transition results in a hard breakdown while the soft breakdown occurs due to a quasi-critical nature of percolation for ultra-thin oxides. In the later part of the thesis, a quantum percolation model is studied in order to explain and model the stress induced leakage current. It is explained that due to the wave nature of electrons, the SILC can be modeled as a tunneling path through the stressed oxide with the smaller tunneling threshold compared to the virgin oxide. In addition to the percolation model, a Markov chain theory is introduced to simulate the movement of electron as a random walk inside the oxide, and the breakdown is simulated using this random-walk of electron through the accumulated traps inside the oxide. It is shown that the trapping-detrapping of electrons results in an electrical noise in the post-breakdown current having 1/f noise characteristics. Using simulation of a resistor network with Markov theory, the conductance of the oxide is computed. An analytical study of a 2-D site percolation system is conducted using recursive methods and useful closed-form expressions are derived for specialized networks.

In the last three decades, the electronic industry has registered a tremendous progress. The continuous and aggressive downsizing of the transistor feature sizes (CMOS scaling) has been the main driver of the astonishing growth and advancement of microelectronic industry. Currently, the CMOS scaling is almost reaching its limits. The gate oxide is now only a few atomic layers thick, and this extremely thin oxide causes a huge leakage current through the oxide. Therefore, a further reduction of the gate oxide thickness is extremely difficult and new materials with higher dielectric constant are being explored. However, the phenomena of oxide breakdown and reliability are still serious issues in these thin oxides. Oxide breakdown exhibits a soft breakdown behavior at low voltages, and this is posing as one of the most crucial reliability issues for scaling of the ultra-thin oxides. In addition, the stress-induced leakage current (SILC) due to oxide has emerged as a scaling problem for the non-volatile memory technologies. In this dissertation, a percolation modeling approach is introduced to study and understand the dramatic changes in the conductivity of a disordered medium. Two different simulation methods of percolative conduction, the site and bond percolation, are studied here. These are used in simulating the post-breakdown conduction inside the oxide. Adopting a Monte-Carlo method, oxide breakdown is modeled using a 2-D percolation theory. The breakdown statistics and post-breakdown characteristics of the oxide are computed using this model. In this work, the effects of different physical parameters, such as dimension and the applied stress are studied. The simulation results show that a thinning of oxide layer and increasing the oxide area result in softening of breakdown. It is observed that the breakdown statistics appear to follow Weibull characteristics. As revealed by simulations, the Weibull slope changes linearly with oxide thickness, while not having a significant change when the area is varied and when the amount of the applied stress is varied. It is shown that the simulation results are well correlated with the experimental data reported in the literature. In this thesis, studying the conduction through the oxide using percolation model, it was discovered that a critical or a quasi-critical phenomenon occurs depending on the oxide dimensions. The criticality of the phase-transition results in a hard breakdown while the soft breakdown occurs due to a quasi-critical nature of percolation for ultra-thin oxides. In the later part of the thesis, a quantum percolation model is studied in order to explain and model the stress induced leakage current. It is explained that due to the wave nature of electrons, the SILC can be modeled as a tunneling path through the stressed oxide with the smaller tunneling threshold compared to the virgin oxide. In addition to the percolation model, a Markov chain theory is introduced to simulate the movement of electron as a random walk inside the oxide, and the breakdown is simulated using this random-walk of electron through the accumulated traps inside the oxide. It is shown that the trapping-detrapping of electrons results in an electrical noise in the post-breakdown current having 1/f noise characteristics. Using simulation of a resistor network with Markov theory, the conductance of the oxide is computed. An analytical study of a 2-D site percolation system is conducted using recursive methods and useful closed-form expressions are derived for specialized networks.

L’émergence de la complexité, et de la vie en particulier, demeure l’une des énigmes les plus complexes pour la science moderne. Des travaux récents ont souligné la pertinence d’un apport de la physique statistique et de la théorie des phénomènes critiques — et en particulier de la théorie des phénomènes à criticalité auto-régulée — relativement à ces champs d’intérêt, tout autant que du rôle des phénomènes de coopération biochimique dans les premiers instants du vivant. La description des mécanismes par lesquels la vie a pu apparaître est par ailleurs d’un intérêt pratique pour l’astrophysique, puisque notre compréhension de ceux-ci module la manière dont l’analyse de biosignatures s’effectue dans le cadre de la recherche de la vie ailleurs dans l’Univers. L’analyse proposée ici porte sur un modèle en théorie des jeux permettant d’étudier les phénomènes de coopération implémenté dans un contexte spatial servant à émuler la dynamique d’un système ayant pu voir apparaître la vie. Une analyse de l’espace des paramètres du modèle révèle que celui-ci affiche des phénomènes de transition de phase et d’auto-organisation de structures spatiales, ces éléments se révélant des adjuvants à l’émergence de la coopération entre joueurs a priori égoïstes, dans un contexte qui à prime abord n’est pas d’emblée favorable à l’apparition de comportements coopératifs. Les résultats obtenus ici semblent supporter que la coopération biochimique puisse apparaître via un phénomène de transition de phase et que le modèle sous-jacent de dilemme du prisonnier itéré sur réseau présenté ici agit comme un système à criticalité autorégulée. / The emergence of complexity, and of life more specifically, is still one of the most intractable conundrums for modern science. Recent work emphasized the relevance of statistical physics and critical phenomena theory’s contribution to those questions — especially of self-organized criticality theory — just as much as the role of biochemical cooperation in life’s first moments. Moreover, the description of the mechanisms by which life could have appeared is of particular interest for astrophysics, because our comprehension of those mechanisms influences how biosignatures are analyzed in the context of the search for life elsewhere in the Universe. The analysis presented here concerns a model in game theory that allows to study cooperation phenomena — implemented in spatial context as to emulate the dynamics of a system in which life could have appeared. An analysis of the model’s parameter space reveals that it displays phase transition and self-organization of spatial structures phenomenon, those elements being adjuvants to the emergence of cooperation between a priori egoist players, in a context that is initially not favorable to the emergence of cooperative behavior. The results obtained here thus seem to support the idea that both biochemical cooperation can emerge through phase transition phenomena, and that the underlying lattice iterated prisoner’s dilemma model used here behaves like a self-organized critical system.