Three applications of fragmentation and Poisson calculus to combinatorics

Abstract : This PhD thesis is devoted to the study of three combinatorial models occurring in probability theory. We are first interested in the heights of fragmentation trees. When the dislocation measure is fixed, two different regimes appear depending on the vertex capacity: above a critical capacity the heights of the trees have the same asymptotics whereas below that critical parameter the trees get significantly higher as long as the rupture threshold decreases. We then present results obtained with Nicolas Curien on the quadtree. We explicit the asymptotic behaviors of the mean costs of the partial match queries. Here again, fragmentation theory plays a key role. Finally, we study large random graphs, critical for the configuration model. We prove that, under some assumptions, the sequences of the component sizes of those graphs, once properly rescaled, converge in a certain sense to a non-trivial random sequence that is caracterized. The situation is very different depending on whether the degree distribution has finite third moment or is a power law distribution with exponent between 3 and 4.
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Mathematics. Université Pierre et Marie Curie - Paris VI, 2011. English
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https://tel.archives-ouvertes.fr/tel-00605743
Contributor : Adrien Joseph <>
Submitted on : Monday, July 4, 2011 - 11:27:37 AM
Last modification on : Monday, July 4, 2011 - 11:40:53 AM

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Adrien Joseph. Three applications of fragmentation and Poisson calculus to combinatorics. Mathematics. Université Pierre et Marie Curie - Paris VI, 2011. English. <tel-00605743>

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