The dual tree of a recursive triangulation of the disk

TL;DR

This paper proves that the dual of a recursively constructed random disk triangulation converges to a limit real tree, confirming a conjecture and using a novel contraction method approach.

Contribution

It introduces a new approach based on the contraction method to prove the convergence of the dual structure in recursive disk triangulations.

Findings

The dual of the recursive lamination converges to a limit real tree.

The convergence is almost sure in the Gromov-Hausdorff sense.

The limit is encoded by the same process as the original triangulation.

Abstract

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.