Recollement aléatoire d'espaces métriques

We construct random metric spaces by gluing together an infinite sequence of pointed metric spaces that we call blocks. At each step, we glue the next block to the structure constructed so far by randomly choosing a point on the structure and then identifying it with the distinguished point of the block. The random object that we study is the completion of the structure that we obtain after an infinite number of steps. We introduce a sequence (w_n)_{n≥1} that we call the weights of the blocks. At each step, the probability that the next block is glued onto any of the preceding blocks is proportional to its weight. We suppose that the blocks are i.i.d. copies of the same random metric space, scaled by deterministic factors that we call (λ_n)n≥1. We work under some conditions on the distribution of the blocks ensuring that they a.s. have dimension d, for some d≥0.The main contribution of this paper is the computation of the Hausdorff dimension of the set L of points which appear during the completion procedure when the sequences (λ_n)_{n≥1} and (w_n)_{n≥1} typically behave like a power of n, say n^{−α} for the scaling factors and n^{−β} for the weights, with α > 0 and β ∈ R. For a large domain of α and β we have dim(L) = 1/α . However for β > 1 and α 0 et β ∈ R. Pour un large domaine de α et β on a dim(L) = 1/α. Cependant, pour β > 1 et α < 1/d, nos résultats révèlent un phénomène intéressant: la dimension de L présente une dépendance non-triviale en α, β et d, à savoir dim (L) = (2β − 1 − 2 (β − 1)(β − αd) )/α. Le calcul de cette dimension dans ce cas faire intervenir de nouveaux outils, spécifiques à ce modèle