Branching processes and Erdős-Rényi graph

Abstract : This thesis is composed by three chapters and its main theme is branching processes.The first chapter is devoted to the study of the Yule tree and the binary search tree. We obtain oscillation results on the expectation, the variance and the distribution of the height of these trees and confirm a Drmota's conjecture. Moreover, the Yule tree can be seen as a particular instance of lattice branching random walk, our results thus allow a better understanding of these processes.In the second chapter, we study the number of particles killed at 0 for a Brownian motion with supercritical drift conditioned to extinction. We finally highlight a new phase transition in terms of the drift for the tail of the distributions of these variables.The main object of the last chapter is the Erdős–Rényi graph in the critical case: $G(n,1/n)$. By using coupling and scaling, we show that, when $n$ grows, the scaling process is asymptotically a coalescence-fragmentation process which acts on real graphs. The coalescent part is of multiplicative type and the fragmentations happen according a certain Poisson point process.
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Submitted on : Monday, April 9, 2018 - 10:35:07 AM
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Pierre-Antoine Corre. Branching processes and Erdős-Rényi graph. Probability [math.PR]. Université Pierre et Marie Curie - Paris VI, 2017. English. ⟨NNT : 2017PA066409⟩. ⟨tel-01761447⟩



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