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Με την βοήθεια τεχνικών για συστήματα αλληλεπιδρώντων σωματιδίων, σκιαγραφήθηκαν και αποδείχθηκαν θεωρήματα για μοντέλα γνώμης και πολιτιστικής δυναμικής. Τα χωρικά αυτά στοχαστικά μοντέλα εξετάζονται ως γενικεύσεις με μια παράμετρο εμπιστοσύνης ε του γνωστού μοντέλου ψηφοφόρου. Το κεντρικό ερώτημα είναι ο καθορισμός της ασυμπτωτικής δυναμικής, η οποία ενδέχεται να εμφανίζει μετάβαση φάσης από μια ποιοτική συμπεριφορά σε κάποια άλλη. Τα παραχθέντα θεωρήματα αφορούν: α) στην επέκταση του θεωρήματος ομαδοποίησης του Lanchier (2012) σε αυθαίρετους γράφους απόψεων, και β) στην εφαρμογή της μεθοδολογίας των Bramson και Griffeath (1989) σε δυο συστήματα με ουδέτερες αλληλεπιδράσεις, την ουδέτερη εκδοχή των κυκλικών συστημάτων σωματιδίων και γ) το μοντέλο Axelrod για την διάχυση των πολιτιστικών περιοχών. Στα δυο τελευταία μοντέλα εξετάζονται τα φαινόμενα τόσο της καθήλωσης (η άποψη κάθε δράστη μεταβάλλεται πεπερασμένα συχνά) όσο και του κατακερματισμού (μη ομαδοποίηση) του άπειρου συστήματος. / By the use of techniques from interacting particle systems, heuristics and proof have been produced for opinion and cultural dynamical models. These stochastic spatial models are investigated as generalizations with a confidence parameter ε of the well-known voter model. The main question is the characterization of dynamics in the asymptotic limit of time, which may exhibit phase transition from one qualitative behavior to another. The produced theorems are: a) an extension of the clustering theorem by Lanchier (2012) to arbitrary opinion graphs, and b) the appropriation of the Bramson and Griffeath (1989) methodology for systems with neutral interactions, namely, a neutral version of cyclic particle systems and c) the model of Axelrod for the diffusion of cultural domains. In the last two models, the studied phenomena is the fixation of the infinite system (each agent changes her opinion finitely often) to a fragmented configuration (non-clustering).

abstract: This dissertation examines six different models in the field of econophysics using interacting particle systems as the basis of exploration. In each model examined, the underlying structure is a graph G = (V , E ), where each x ∈ V represents an individual who is characterized by the number of coins in her possession at time t. At each time step t, an edge (x, y) ∈ E is chosen at random, resulting in an exchange of coins between individuals x and y according to the rules of the model. Random variables ξt, and ξt(x) keep track of the current configuration and number of coins individual x has at time t respectively. Of particular interest is the distribution of coins in the long run. Considered first are the uniform reshuffling model, immediate exchange model and model with saving propensity. For each of these models, the number of coins an individual can have is nonnegative and the total number of coins in the system is conserved for all time. It is shown here that the distribution of coins converges to the exponential distribution, gamma distribution and a pseudo gamma distribution respectively. The next two models introduce debt, however, the total number of coins again remains fixed. It is shown here that when there is an individual debt limit, the number of coins per individual converges to a shifted exponential distribution. Alternatively, when a collective debt limit is imposed on the whole population, a heuristic argument is given supporting the conjecture that the distribution of coins converges to an asymmetric Laplace distribution. The final model considered focuses on the effect of cooperation on a population. Unlike the previous models discussed here, the total number of coins in the system at any given time is not bounded and the process evolves in continuous time rather than in discrete time. For this model, death of an individual will occur if they run out of coins. It is shown here that the survival probability for the population is impacted by the level of cooperation along with how productive the population is as whole. / Dissertation/Thesis / Doctoral Dissertation Mathematics 2019

This thesis is concerned with certain properties of stochastic growth models. A stochastic growth model is a model of infection spread, through a population of individuals, that incorporates an element of randomness. The models we consider are variations on the contact process, the simplest stochastic growth model with a recurrent infection. Three main examples are considered. The first example is a version of the contact process on the complete graph that incorporates dynamic monogamous partnerships. To our knowledge, this is the first rigorous study of a stochastic spatial model of infection spread that incorporates some form of social dynamics. The second example is a non-monotonic variation on the contact process, taking place on the one-dimensional lattice, in which there is a random incubation time for the infection. Some techniques exist for studying non-monotonic particle systems, specifically models of competing populations [38] [12]. However, ours is the first rigorous study of a non-monotonic stochastic spatial model of infection spread. The third example is an additive two-stage contact process, together with a general duality theory for multi-type additive growth models. The two-stage contact process is first introduced in \cite{krone}, and several open questions are posed, most of which we have answered. There are many examples of additive growth models in the literature [26] [16] [29] [49], and most include a proof of existence of a dual process, although up to this point no general duality theory existed. In each case there are three main goals. The first is to identify a phase transition with a sharp threshold or ``critical value'' of the transmission rate, or a critical surface if there are multiple parameters. The second is to characterize either the invariant measures if the population is infinite, or to characterize the metastable behaviour and the time to extinction of the disease, if the population is finite. The final goal is to determine the asymptotic behaviour of the model, in terms of the invariant measures or the metastable states. In every model considered, we identify the phase transition. In the first and third examples we show the threshold is sharp, and in the first example we calculate the critical value as a rational function of the parameters. In the second example we cannot establish sharpness due to the lack of monotonicity. However, we show there is a phase transition within a range of transmission rates that is uniformly bounded away from zero and infinity, with respect to the incubation time. For the partnership model, we show that below the critical value, the disease dies out within C log N time for some C>0, where N is the population size. Moreover we show that above the critical value, there is a unique metastable proportion of infectious individuals that persists for at least e^{\gamma N}$ time for some $\gamma>0$. For the incubation time model, we use a block construction, with a carefully chosen good event to circumvent the lack of monotonicity, in order to show the existence of a phase transition. This technique also guarantees the existence of a non-trivial invariant measure. Due to the lack of additivity, the identification of all the invariant measures is not feasible. However, we are able to show the following is true. By rescaling time so that the average incubation period is constant, we obtain a limiting process as the incubation time tends to infinity, with a sharp phase transition and a well-defined critical value. We can then show that as the incubation time approaches infinity (or zero), the location of the phase transition in the original model converges to the critical value of the limiting process (respectively, the contact process). For the two-stage contact process, we can show that there are at most two extremal invariant measures: the trivial one, and a non-trivial upper invariant measure that appears above the critical value. This is achieved using known techniques for the contact process. We can show complete convergence, from any initial configuration, to a combination of these measures that is given by the survival probability. This, and some additional results, are in response to the questions posed by Krone in his original paper \cite{krone} on the model. We then generalize these ideas to develop a theory of additive growth models. In particular, we show that any additive growth model, having any number of types and interactions, will always have a dual process that is also an additive growth model. Under the additional technical condition that the model preserves positive correlations, we can then harness existing techniques to conclude existence of at most two extremal invariant measures, as well as complete convergence. / Graduate

Simulation and game content includes the levels, models, textures, items, and other objects encountered and possessed by players during the game. In most modern video games and simulation software, the set of content shipped with the product is static and unchanging, or at best, randomized within a narrow set of parameters. However, ideally, if game content could be constantly and automatically renewed, players would remain engaged longer in the evolving stream of content. This dissertation introduces three novel technologies that together realize this ambition. (1) The first, NEAT Particles, is an evolutionary method to enable users to quickly and easily create complex particle effects through a simple interactive evolutionary computation (IEC) interface. That way, particle effects become an evolvable class of content, which is exploited in the remainder of the dissertation. In particular, (2) a new algorithm called content-generating NeuroEvolution of Augmenting Topologies (cgNEAT) is introduced that automatically generates graphical and game content while the game is played, based on the past preferences of the players. Through cgNEAT, the game platform on its own can generate novel content that is designed to satisfy its players. Finally, (3) the Galactic Arms Race (GAR) multiplayer online video game is constructed to demonstrate these techniques working on a real online gaming platform. In GAR, which was made available to the public and playable online, players pilot space ships and fight enemies to acquire unique particle system weapons that are automatically evolved by the cgNEAT algorithm. The resulting study shows that cgNEAT indeed enables players to discover a wide variety of appealing content that is not only novel, but also based on and extended from previous content that they preferred in the past. The implication is that with cgNEAT it is now possible to create applications that generate their own content to satisfy users, potentially significantly reducing the cost of content creation and considerably increasing entertainment value with a constant stream of evolving content.

We explore the effect of multiscale structure on weakly interacting diffusions through two main projects. In the first, we consider a collection of weakly interacting diffusion processes moving in a two-scale locally periodic environment. We study the large deviations principle of the empirical distribution of the particles' positions in the combined limit as the number of particles grow to infinity and the time-scale separation parameter goes to zero simultaneously. We make use of weak convergence methods providing a convenient representation for the large deviations rate function, which allow us to characterize the effective controlled mean field dynamics. In addition, we obtain equivalent representations for the large deviations rate function of the form of Dawson-G\"artner which hold even in the case where the diffusion matrix depends on the empirical measure and when the particles undergo averaging in addition to the propagation of chaos. In the second, we consider a fully-coupled slow-fast system of McKean-Vlasov SDEs with full dependence on the slow and fast component and on the law of the slow component and derive convergence rates to its homogenized limit. We do not make periodicity assumptions, but we impose conditions on the fast motion to guarantee ergodicity. In the course of the proof we obtain related ergodic theorems and we gain results on the regularity of Poisson type of equations and of the associated Cauchy-Problem on the Wasserstein space that are of independent interest.

Particle systems are used to create visual effects in real-time applications such as computer games. However, emitted particles are often transient and do not leave a lasting impact on a 3D scene. This thesis work presents a real-time method that enables GPU particle systems to paint meshes in a 3D scene as the result of particle collisions, thus adding detail to and leaving a lasting impact on a scene. The method uses screen space collision detection and a mapping from screen space to texture space of meshes to determine where to apply paint. The method was tested for its time complexity and how well it performed in scenarios similar to those found in computer games. The results shows that the method probably can be used in computer games. Performance and visual fidelity of the paint application is not directly dependent on the amount of simulated particles, but depends only on the complexity of the meshes and their texture mapping as wellas the resolution of the paint. It is concluded that the method is renderer agnostic and could be added to existing GPU particle systems and that other types of effects than those showed in the thesis could be achieved by using the method.

This thesis is divided into two parts: In the first one (Chapters 1 and 2), we deal with problems arising from quantitative homogenization of the random elliptic operator in divergence form $-\\nabla \\cdot a \\nabla$. In Chapter 1 we study existence and stochastic bounds for the Green function $G$ associated to $-\\nabla \\cdot a \\nabla$ in the case of systems. Without assuming any regularity on the coefficient field $a= a(x)$, we prove that for every (measurable) uniformly elliptic tensor field $a$ and for almost every point $y \\in \\mathbb^d$, there exists a unique Green\'s function centred in $y$ associated to the vectorial operator $-\\nabla \\cdot a\\nabla $ in $\\mathbb{R}^d$, $d> 2$. In addition, we prove that if we introduce a shift-invariant ensemble $\\langle\\cdot \\rangle$ over the set of uniformly elliptic tensor fields, then $\\nabla G$ and its mixed derivatives $\\nabla \\nabla G$ satisfy optimal pointwise $L^1$-bounds in probability. Chapter 2 deals with the homogenization of $-\\nabla \\cdot a \\nabla$ to $-\\nabla \\ah \\nabla$ in the sense that we study the large-scale behaviour of $a$-harmonic functions in exterior domains $\\{ |x| > r \\}$ by comparing them with functions which are $\\ah$-harmonic. More precisely, we make use of the first and second-order correctors to compare an $a$-harmonic function $u$ to the two-scale expansion of suitable $\\ah$-harmonic function $u_h$. We show that there is a direct correspondence between the rate of the sublinear growth of the correctors and the smallness of the relative homogenization error $u- u_h$. The theory of stochastic homogenization of elliptic operators admits an equivalent probabilistic counterpart, which follows from the link between parabolic equations with elliptic operators in divergence form and random walks. This allows to reformulate the problem of homogenization in terms of invariance principle for random walks. The second part of thesis (Chapters 3 and 4) focusses on this interplay between probabilistic and analytic approaches and aims at exploiting it to study invariance principles in the case of degenerate random conductance models and systems of interacting particles. In Chapter 3 we study a random conductance model where we assume that the conductances are independent, stationary and bounded from above but not uniformly away from $0$. We give a simple necessary and sufficient condition for the relaxation of the environment seen by the particle to be diffusive in the sense of every polynomial moment. As a consequence, we derive polynomial moment estimates on the corrector which imply that the discrete elliptic operator homogenises or, equivalently, that the random conductance model satisfies a quenched invariance principle. In Chapter 4 we turn to a more complicated model, namely the symmetric exclusion process. We show a diffusive upper bound on the transition probability of a tagged particle in this process. The proof relies on optimal spectral gap estimates for the dynamics in finite volume, which are of independent interest. We also show off-diagonal estimates of Carne-Varopoulos type.

Nosso objeto de estudo são os sistemas de spins com interações de longo alcance; em particular, estamos interessados em sistemas cuja probabilidade invariante é o modelo de Ising em A^S, onde A = {-1, 1} é o espaço de spins e S = Z^d é o espaço de sítios. Apresentamos dois resultados originais que são consequências da aplicação de algoritmos de simulação perfeita e de acoplamento no contexto da construção deste tipo de sistemas e de suas respectivas probabilidades invariantes. / Our object of interest are spin systems with long-range interactions. As a special case, we are interested in systems whose invariant measure is the Ising model on A^S, where A = {-1, 1} is the space of spins and S = Z^d is the space of sites. We present two original results that are byproducts of the application of Perfect Simulation and Coupling algorithms in the context of the construction of these spin systems and their respective invariant measures.

Nesta tese consideramos uma nova classe de sistemas markovianos de partículas com infinitas componentes interagentes. O sistema representa a evolução temporal dos potenciais de membrana de um conjunto infinito de neurônios interagentes. Provamos a existência e unicidade do processo construindo um pseudo-algoritmo de simulação perfeita e mostrando que este algoritmo roda em um número finito de passos quase certamente. Estudamos também o comportamento do sistema quando consideramos apenas um conjunto finito de neurônios. Neste caso, construímos um procedimento de simulação perfeita para o acoplamento entre o processo limitado a um conjunto finito de neurônios e o processo que considera todos os neurônios do sistema. Como consequência encontramos um limitante superior para a probabilidade de discrepância entre os processos. / We consider a new class of interacting particle systems with a countable number of interacting components. The system represents the time evolution of the membrane potentials of an infinite set of interacting neurons. We prove the existence and uniqueness of the process, by the construction of a perfect simulation procedure. We show that this algorithm is successful, that is, we show that the number of steps of the algorithm is finite almost surely. We also study the behaviour of the system when we consider only a finite number of neurons. In this case, we construct a perfect simulation procedure for the coupling of the process with a finite number of neurons and the process with a infinite number of neurons. As a consequence we obtain an upper bound for the error we make when sampling from a finite set of neurons instead of the infinite set of neurons.