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Register a new userMass-structure of weighted real trees
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Noah Forman
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Rooted, weighted continuum random trees are used to describe limits of sequences of random discrete trees. Formally, they are random quadruples $(mathcal{T},d,r,p)$, where $(mathcal{T},d)$ is a tree-like metric space, $rinmathcal{T}$ is a distinguished root, and $p$ is a probability measure on this space. The underlying branching structure is carried implicitly in the metric $d$. We explore various ways of describing the interaction between branching structure and mass in $(mathcal{T},d,r,p)$ in a way that depends on $d$ only by way of this branching structure. We introduce a notion of mass-structure equivalence and show that two rooted, weighted $mathbb{R}$-trees are equivalent in this sense if and only if the discrete hierarchies derived by i.i.d. sampling from their weights, in a manner analogous to Kingmans paintbox, have the same distribution. We introduce a family of trees, called interval partition trees that serve as representatives of mass-structure equivalence classes, and which naturally represent the laws of the aforementioned hierarchies.
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We consider fixed-point equations for probability measures charging measured compact metric spaces that naturally yield continuum random trees. On the one hand, we study the existence/uniqueness of the fixed-points and the convergence of the corresponding iterative schemes. On the other hand, we study the geometric properties of the random measured real trees that are fixed-points, in particular their fractal properties. We obtain bounds on the Minkowski and Hausdorff dimension, that are proved tight in a number of applications, including the very classical continuum random tree, but also for the dual trees of random recursive triangulations of the disk introduced by Curien and Le Gall [Ann Probab, vol. 39, 2011]. The method happens to be especially efficient to treat cases for which the mass measure on the real tree induced by natural encodings only provides weak estimates on the Hausdorff dimension.
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