Scaling limit of random planar quadrangulations with a boundary

Bettinelli, Jérémie

Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015) no. 2, pp. 432-477.

On s’intéresse à la limite d’échelle de grandes quadrangulations planaires à bord dont la longueur du bord est de l’ordre de la racine carrée du nombre de faces. On considère une suite $\left({\sigma }_n\right)$ d’entiers telle que ${\sigma }_n/\sqrt{2n}$ tende vers un certain $\sigma \in [0,\infty ]$. Pour tout $n\ge 1$, on note ${𝔮}_n$ une carte aléatoire uniformément distribuée dans l’ensemble des quadrangulations planaires enracinées à bord ayant $n$ faces internes et $2{\sigma }_n$ demi-arêtes sur le bord. Dans le cas où $\sigma \in (0,\infty )$, on voit ${𝔮}_n$ comme un espace métrique en munissant l’ensemble de ses sommets de la distance de graphe, renormalisée par le facteur $n^{-1/4}$. On montre que cet espace métrique converge en loi, tout du moins le long d’une sous-suite, vers un espace métrique limite aléatoire, au sens de la topologie de Gromov–Hausdorff. On montre que l’espace métrique limite est presque sûrement un espace de dimension de Hausdorff $4$ ayant un bord de dimension $2$ qui est homéomorphe au disque de dimension $2$. Pour $\sigma =0$, on a également la même convergence mais cette fois-ci, l’extraction d’une sous-suite n’est plus nécessaire et la limite est l’espace métrique connu sous le nom de carte brownienne. Pour $\sigma =\infty$, le bon facteur d’échelle devient ${\sigma }_n^{-1/2}$ et on a convergence vers l’arbre continu brownien d’Aldous.

We discuss the scaling limit of large planar quadrangulations with a boundary whose length is of order the square root of the number of faces. We consider a sequence $\left({\sigma }_n\right)$ of integers such that ${\sigma }_n/\sqrt{2n}$ tends to some $\sigma \in [0,\infty ]$. For every $n\ge 1$, we denote by ${𝔮}_n$ a random map uniformly distributed over the set of all rooted planar quadrangulations with a boundary having $n$ faces and $2{\sigma }_n$ half-edges on the boundary. For $\sigma \in (0,\infty )$, we view ${𝔮}_n$ as a metric space by endowing its set of vertices with the graph metric, rescaled by $n^{-1/4}$. We show that this metric space converges in distribution, at least along some subsequence, toward a limiting random metric space, in the sense of the Gromov–Hausdorff topology. We show that the limiting metric space is almost surely a space of Hausdorff dimension $4$ with a boundary of Hausdorff dimension $2$ that is homeomorphic to the two-dimensional disc. For $\sigma =0$, the same convergence holds without extraction and the limit is the so-called Brownian map. For $\sigma =\infty$, the proper scaling becomes ${\sigma }_n^{-1/2}$ and we obtain a convergence toward Aldous’s CRT.

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