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Mouvement brownien branchant et autres modèles hiérarchiques en physique statistique

Abstract : Branching Brownian motion (BBM) is a particle system, where particles move and reproduce randomly. Firstly, we study precisely the phase transition occuring for this particle system close to its minimum, in the setting of the so-called near-critical case. Then, we describe the universal 1-stable fluctuations appearing in the front of BBM and identify the typical behavior of particles contributing to them. A version of BBM with selection, where particles are killed when going down at a distance larger than L from the highest particle, is also sudied: we see how this selection rule affects the speed of the fastest individuals in the population, when L is large. Thereafter, motivated by temperature chaos in spin glasses, we study the 2-dimensional discrete Gaussian free field, which is a model with an approximative hierarchical structure and properties similar to BBM, and show that, from this perspective, it behaves differently than the Random Energy Model. Finally, the last part of this thesis is dedicated to the Derrida-Retaux model, which is also defined by a hierarchical structure. We introduce a continuous time version of this model and exhibit a family of exactly solvable solutions, which allows us to answer several conjectures stated on the discrete time model.
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Submitted on : Monday, February 15, 2021 - 10:34:22 AM
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  • HAL Id : tel-02435953, version 2

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Michel Pain. Mouvement brownien branchant et autres modèles hiérarchiques en physique statistique. Probabilités [math.PR]. Sorbonne Université, 2019. Français. ⟨NNT : 2019SORUS305⟩. ⟨tel-02435953v2⟩

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