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Scaling Limit of Random Planar Quadrangulations with a Boundary

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 $(σ_n)$ of integers such that $σ_n/\sqrt{2n}$ tends to some $σ\in[0,\infty]$. For every $n \ge 1$, we call $q_n$ a random map uniformly distributed over the set of all rooted planar quadrangulations with a boundary having $n$ faces and $2σ_n$ half-edges on the boundary. For $σ\in (0,\infty)$, 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 $σ=0$, the same convergence holds without extraction and the limit is the so-called Brownian map. For $σ=\infty$, the proper scaling becomes $σ_n^{-1/2}$ and we obtain a convergence toward Aldous's CRT.