Random stable looptrees
By Nicolas Curien, Igor Kortchemski. 2013
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We introduce a class of random compact metric spaces $\mathscr{L}_{\alpha}$ indexed by $\alpha~\in(1,2)$ and which we call stable looptrees. They are made of a collection of random loops glued together along a tree structure, and can be informally be viewed as dual graphs of $\alpha$-stable Levy trees. We study their properties and prove in particular that the Hausdorff dimension of $ \mathscr{L}_{\alpha}$ is almost surely equal to $\alpha$. We also show that stable looptrees are universal scaling limits, for the Gromov-Hausdorff topology, of various combinatorial models. In a companion paper, we prove that the stable looptree of parameter $ \frac{3}{2}$ is the scaling limit of cluster boundaries in critical site-percolation on large random triangulations.
Subfields: Computational Theory and Mathematics, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/26ee9a929e6a4a384203c55d593264fdd4205d92
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In Computational Theory and Mathematics
In Mathematical Physics
Citations in the corpus
- Sub-exponential tail bounds for conditioned stable Bienaymé–Galton–Watson trees (2015, background): “Bienaymé–Galton–Watson trees belonging to domains of attraction of stable laws have recently appeared in a number of two-dimensional statistical physics models, in connection with scaling limits of random maps with large faces [34], scaling limits of critical site-percolation clusters on infinite ra”