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

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-Gartner 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.

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.

We study the totally asymmetric exclusion process on the positive integers with a single particle source at the origin. Liggett (1975) has shown that the long term behaviour of this process has a phase transition: If the particle production rate at the source and the initial density are below certain critical values, the stationary measure is a product measure, otherwise the stationary measure is spatially correlated. Following the approach of Derrida et al. (1993) it was shown by Grosskinsky (2004) that these correlations can be described by means of a matrix product representation. In this thesis we derive a large deviation principle with explicit rate function for the particle density in a macroscopic box based on this representation. The novel and rigorous technique we develop for this problem combines spectral theoretical and combinatorial ideas and has the potential to be applicable to other models described by matrix products.

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.

We first introduce some coupling of a finite number of Probabilistic Cellular Automata dynamics (PCA), preserving the stochastic ordering. Using this tool, for a general attractive probabilistic cellular automata on SZd, where S is finite, we prove that a condition (A) is equivalent to the (time-) convergence towards equilibrium of this Markovian parallel dynamics, in the uniform norm, exponentially fast. This condition (A) means the exponential decay of the influence from the boundary for the invariant measures of the system restricted to finite ‘box’-volume. For a class of reversible PCA dynamics on {−1, +1}Zd / with a naturally associated Gibbsian potential ϕ, we prove that a Weak Mixing condition for ϕ implies the validity of the assumption (A); thus the ‘exponential ergodicity’ of the dynamics towards the unique Gibbs measure associated to ϕ holds. On some particular examples of this PCA class, we verify that our assumption (A) is weaker than the Dobrushin-Vasershtein ergodicity condition. For some special PCA, the ‘exponential ergodicity’ holds as soon as there is no phase transition.

We study a stochastic particle system with a logarithmically-singular inter-particle interaction potential which allows for inelastic particle collisions. We relate the squared Bessel process to the evolution of localized clusters of particles, and develop a numerical method capable of detecting collisions of many point particles without the use of pairwise computations, or very refined adaptive timestepping. We show that when the system is in an appropriate parameter regime, the hydrodynamic limit of the empirical mass density of the system is a solution to a nonlinear Fokker-Planck equation, such as the Patlak-Keller-Segel (PKS) model, or its multispecies variant. We then show that the presented numerical method is well-suited for the simulation of the formation of finite-time singularities in the PKS, as well as PKS pre- and post-blow-up dynamics. Additionally, we present numerical evidence that blow-up with an increasing total second moment in the two species Keller-Segel system occurs with a linearly increasing second moment in one component, and a linearly decreasing second moment in the other component.