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 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 T, which is encoded by M. This confirms a conjecture of Curien and Le Gall.