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Arbres couvrants minimums aléatoires inhomogènes, propriétés et limite

Abstract : In this thesis we study a specific type of inhomogeneous random minimum spanning trees and their related graphs. For large positive integers n, we put positive weights (wi)i≤n on the nodes of the complete graph of size n. Moreover, the weights (wi)i≤n that we consider in this thesis verify a finite third moment condition. This condition is natural given the history of those trees. Then, we give to each edge {i,j} of the complete graph an edge capacity which is an exponential random variable of parameter wiwj independently of everything else. We then build the minimum spanning tree of the complete graph with these edge capacities. In Chapter 2 we prove asymptotic properties for rank-1 inhomogeneous random graphs in the so-called barely super-critical regime. The novelty of this chapter lies in the detailed study of those graphs, and on proofs of original concentration inequalities for sampling without replacement. In Chapter 3, we use the results of Chapter 2 in order to prove that the expectation of the typical distances and of the diameter of our minimum spanning trees is of order n(1/3) when n is large. As a corollary of our work, we answer a conjecture from statistical physics regarding typical distances in a model of minimum spanning trees closely related to our model. Finally, in Chapter 4, we prove that our minimum spanning trees, seen as metric spaces and with distances rescaled by n(1/3) converge in distribution to a non-trivial compact metric space for the Gromov-Hausdorff topology.
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Othmane Safsafi. Arbres couvrants minimums aléatoires inhomogènes, propriétés et limite. Topologie algébrique [math.AT]. Sorbonne Université, 2021. Français. ⟨NNT : 2021SORUS201⟩. ⟨tel-03457422⟩

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