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Combinatoire analytique et modèles d'urnes

Basile Morcrette 1, 2
2 APR - Algorithmes, Programmes et Résolution
LIP6 - Laboratoire d'Informatique de Paris 6
Abstract : This thesis studies Pólya urns through the analytic combinatorics point of view. Urns are conceptually very simple models of growth or extinction dynamics for which the limiting behaviors are extremely diverse. Those models are widely studied by probabilistic approach, but the precise understanding of the variety of limit laws is still an open question. Since 2005, the work of Flajolet et al. shows that an analytic combinatorics approach can be very fruitful for those questions: the study of the properties (nature, singularities) of generating functions linked to urns provides access to many precisions on limit laws. This thesis is a continuation of this work. First, the determination of the nature of the generating functions of urns by a high tech algorithm of computer algebra (au- tomatic Guess'n'Prove) identifies which functions are algebraic. Then, we lead exact and asymptotic analysis for algebraic classes and precise properties on limiting behaviors are thus derived (moments structure, rate of convergence, local limit properties). Second, a study of some non algebraic urns is done through concrete examples linked to some models of social networks, or the combinatorics of some boolean formulas. Third, through the extension of classical models (unbalanced models, random entries for substitution rules), we show that the symbolic aspects of analytic combinatorics are thriving. More specifically, a general combinatorial study for non necessarily balanced urns is done for the first time and links any urn to a partial differential equation.
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Submitted on : Wednesday, July 10, 2013 - 11:41:09 AM
Last modification on : Friday, January 8, 2021 - 5:32:06 PM
Long-term archiving on: : Monday, October 14, 2013 - 10:35:59 AM


  • HAL Id : tel-00843046, version 1


Basile Morcrette. Combinatoire analytique et modèles d'urnes. Combinatoire [math.CO]. Université Pierre et Marie Curie - Paris VI, 2013. Français. ⟨tel-00843046⟩



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