Le processus de contact sur le graphe Booléen

Abstract : This thesis is a contribution to the mathematical study of interacting particle systems, and more precisely of random growth models representing a spreading shape over time in a lattice. These processes occur when one is interested in the evolution of a population or the spread of an epidemic. One of those models is the contact process introduced by Harris in 1974 with the goal of representing this specific spread. It is one of the simplest interacting particle systems that exhibits a critical phenomenon and today, in the cubic lattice, its behavior is well-known on each phase. Here, we study the standard contact process the Boolean graph which is a random graph in Rd. Our work in particular was motivated by the following result of L. M´enard and A. Singh: on this random network, the contact process admits a non-trivial phase transition. This is the first example of a non-bounded degree graph on which the phase transition of the contact process is non-trivial. We begin our work with a study of the supercritical Boolean model to find large scale regularity properties that allow us to adapt the usual approaches of the study of the contact process on the deterministic networks to the Boolean random framework. In our main result, we show that our model satisfies a deterministic asymptotic shape theorem. In fact, it appears that the large scale regularity properties mentioned above are sufficient to obtain an asymptotic shape theorem on other random graphs
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Tom Riblet. Le processus de contact sur le graphe Booléen. Mathématiques [math]. Université de Lorraine, 2019. Français. ⟨NNT : 2019LORR0007⟩. ⟨tel-02096425⟩

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