The dual tree of a recursive triangulation of the disk

In the recursive lamination of the disk, one tries to add chords one after another at random; a chord is kept and inserted if it does not intersect any of the previously inserted ones. Curien and Le Gall [Ann. Probab. 39 (2011) 2224-2270] have proved that the set of chords converges to a limit triangulation of the disk encoded by a continuous process $\mathscr{M}$. Based on a new approach resembling ideas from the so-called contraction method in function spaces, we prove that, when properly rescaled, the planar dual of the discrete lamination converges almost surely in the Gromov-Hausdorff sense to a limit real tree $\mathscr{T}$, which is encoded by $\mathscr{M}$. This confirms a conjecture of Curien and Le Gall.

💡 Research Summary

The paper investigates the geometric limit of the planar dual tree that arises from the recursive lamination (or triangulation) of the unit disk. In the recursive lamination model, chords are added one by one at random; a newly drawn chord is kept only if it does not intersect any chord that has already been inserted. Curien and Le Gall (Ann. Probab. 39 (2011) 2224‑2270) showed that the random set of chords converges, as the number of steps tends to infinity, to a random triangulation of the disk that can be encoded by a continuous stochastic process 𝓜 (often called the “Brownian triangulation” or “lamination process”). While the limiting triangulation itself is well understood, the behavior of its planar dual—i.e., the tree whose vertices correspond to the faces (triangles) of the triangulation and whose edges connect adjacent faces—remained conjectural. Curien and Le Gall conjectured that, after an appropriate rescaling, the dual tree should converge in the Gromov–Hausdorff sense to a random real tree 𝓣 that is also encoded by the same process 𝓜.

The authors answer this conjecture affirmatively by introducing a novel analytical framework that adapts the contraction method, traditionally used for recursive distributional equations in real‑valued settings, to function spaces. Their approach proceeds through several key steps:

1. Encoding the discrete dual tree.
   For each integer n, let Tₙ denote the planar dual of the lamination after n chords have been inserted. The authors equip Tₙ with the graph distance and then rescale distances by n⁻¹ᐟ², which is the natural scaling for objects that converge to a continuum random tree. This yields a sequence of metric spaces (Tₙ, dₙ) that lives in the space of compact metric spaces equipped with the Gromov–Hausdorff metric.

2. Recursive decomposition.
   The insertion of a new chord either splits an existing face into two or leaves the current configuration unchanged. This operation translates into a recursive decomposition of the dual tree: the tree can be viewed as the concatenation of two independent copies of a smaller dual tree, each rescaled by a random factor that depends on the size of the two sub‑faces created by the chord. This yields a stochastic fixed‑point equation of the form
   \