Random gluing of metric spaces

Abstract

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. In (Ann. Inst. Fourier (Grenoble) 67 (2017) 1963–2001), Curien and Haas study the case of segments, where the sequence of lengths is deterministic and typically behaves like $n^{-\alpha}$. They proved that for $\alpha>0$, the resulting tree is compact and that the Hausdorff dimension of its set of leaves is $\alpha^{-1}$. The aim of this paper is to handle a much more general case in which the blocks are i.i.d. copies of the same random metric space, scaled by deterministic factors that we call $(\lambda_{n})_{n\geq1}$. We work under some conditions on the distribution of the blocks ensuring that their Hausdorff dimension is almost surely $d$, for some $d\geq0$. We also introduce a sequence $(w_{n})_{n\geq1}$ 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. The main contribution of this paper is the computation of the Hausdorff dimension of the set $\mathcal{L}$ of points which appear during the completion procedure when the sequences $(\lambda_{n})_{n\geq1}$ and $(w_{n})_{n\geq1}$ typically behave like a power of $n$, say $n^{-\alpha}$ for the scaling factors and $n^{-\beta}$ for the weights, with $\alpha>0$ and $\beta\in\mathbb{R}$. For a large domain of $\alpha$ and $\beta$, we have the same behaviour as the one observed in (Ann. Inst. Fourier (Grenoble) 67 (2017) 1963–2001), which is that $\operatorname{dim}_{\mathrm{H}}(\mathcal{L})=\alpha^{-1}$. However, for $\beta>1$ and $\alpha<1/d$, our results reveal an interesting phenomenon: the dimension has a nontrivial dependence in $\alpha$, $\beta$ and $d$, namely \begin{equation*}\operatorname{dim}_{\mathrm{H}}(\mathcal{L})=\frac{2\beta-1-2\sqrt{(\beta-1)(\beta-\alpha d)}}{\alpha}.\end{equation*} The computation of the dimension in the latter case involves new tools, which are specific to our model.