The brownian cactus I. Scaling limits of discrete cactuses

Curien, Nicolas; Le Gall, Jean-François; Miermont, Grégory

Annales de l'I.H.P. Probabilités et statistiques, Tome 49 (2013) no. 2, pp. 340-373.

Le cactus d’un graphe pointé est un certain arbre discret associé à ce graphe. De façon similaire, à tout espace métrique géodésique pointé $E$, on peut associer un $ℝ$-arbre appelé cactus continu de $E$. Sous des hypothèses générales, nous montrons que le cactus de cartes planaires aléatoires - dont la loi est déterminée par des poids de Boltzmann, et qui sont conditionnées à avoir un grand nombre fixé de sommets - converge en loi vers un espace limite appelé cactus brownien, au sens de la topologie de Gromov-Hausdorff. De plus, le cactus brownien peut être interprété comme le cactus continu de la carte brownienne.

The cactus of a pointed graph is a discrete tree associated with this graph. Similarly, with every pointed geodesic metric space $E$, one can associate an $ℝ$-tree called the continuous cactus of $E$. We prove under general assumptions that the cactus of random planar maps distributed according to Boltzmann weights and conditioned to have a fixed large number of vertices converges in distribution to a limiting space called the Brownian cactus, in the Gromov-Hausdorff sense. Moreover, the Brownian cactus can be interpreted as the continuous cactus of the so-called Brownian map.

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