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Random tree-like structures : gluings of metric spaces and stable graphs

Abstract : The subject of this thesis is the study of some random metric spaces with a tree-like structure. We first study a construction in which we glue a sequence of metric spaces onto each other in a sequential manner. Under some conditions on the spaces that we aggregate, we compute the Hausdorff dimension of the obtained structure and it has a surprising expression ! We then investigate some asymptotic properties (degrees, height,profile) of two models of growing discrete trees, the weighted recursive trees and the preferential attachment trees with additive fitnesses. The former encodes the underlying discrete structure in the construction described above and the latter have a similar interpretation for some models of discrete growing graphs. We make use of this connection in order to prove scaling limit results for these random discrete graphs towards continuous metric space constructed by a gluing procedure. Last, in a joint work with Christina Goldschmidt and Bénédicte Haas, we investigate the behaviour of the alpha­stable component with fixed surplus. This random metric space appears as the scaling limit of large connected components of the configuration model with heavy-tailed degrees. This abject is almost a tree except for a finite number of cycles. We compute the distribution of the cyclic structure and give a description of the whole space as trees glued along this structure.
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Submitted on : Thursday, October 1, 2020 - 5:32:08 PM
Last modification on : Tuesday, October 20, 2020 - 3:56:19 PM


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  • HAL Id : tel-02955347, version 1


Delphin Sénizergues. Random tree-like structures : gluings of metric spaces and stable graphs. Geometric Topology [math.GT]. Université Sorbonne Paris Cité, 2019. English. ⟨NNT : 2019USPCD013⟩. ⟨tel-02955347⟩



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