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/√2n tends to some σ ∈ [0, ∞]. For every n ≥ 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 2σ_n half-edges on the boundary. For σ ∈ (0, ∞), 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 = ∞, the proper scaling becomes σ_n^-1/2 and we obtain a convergence toward Aldous's CRT.