Random stable looptrees

Abstract

We introduce a class of random compact metric spaces 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 L\'evy trees. We study their properties and prove in particular that the Hausdorff dimension of 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 3/2 is the scaling limit of cluster boundaries in critical site-percolation on large random triangulations.Comment: 35 pages, 12 figures. Final version, Electron. J. Probab. 19 (2014), no. 108, 1-3

Percolation on random triangulations and stable looptrees

“…In particular, we establish that the scaling limit of ∂H • a c conditioned to be large, appropriately rescaled, is the stable looptree of parameter 3/2 introduced in [13], whose definition we now recall.…”

“…1)). In the critical case, this limit is shown to be L 3/2 , the stable looptree of parameter 3/2 introduced in [13]. Our method is based on a surgery technique inspired from Borot, Bouttier and Guitter and on a tree decomposition of triangulations with non-simple boundary.…”

“…They are constructed in [13] using stable processes with no negative jumps, but can also be defined as scaling limits of discrete objects: With every rooted oriented tree (or plane tree) τ , we associate a graph, called the discrete looptree of τ and denoted by Loop(τ ), which is the graph on the set of vertices of τ such that two vertices u and v are joined by an edge if and only if one of the following three conditions are satisfied in τ : u and v are consecutive siblings of a same parent, or u is the first sibling (in the lexicographical order) of v, or u is the last sibling of v, see Fig. 3.…”

“…3. Note that in [13], Loop(τ ) is defined as a different graph, and that here Loop(τ ) is the graph which is denoted by Loop (τ ) in [13]. We view Loop(τ ) as a compact metric space by endowing its vertices with the graph distance (every edge has unit length).…”

“…Now let τ n be a Galton-Watson tree conditioned on having n vertices, whose offspring distribution μ is critical and satisfies μ k ∼ c · k −1−α as k → ∞ for a certain c > 0. In [13,Section 4.2], it is shown that there exists a random compact metric space L α , called the stable looptree of index α, such that where the convergence holds in distribution for the Gromov-Hausdorff topology and where c · M stands for the metric space obtained from M by multiplying all distances by c > 0. Recall that the Gromov-Hausdorff topology gives a sense to the convergence of (isometry classes) of compact metric spaces, see Sect.…”