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 (σn) of integers such that σn/ √ 2n tends to some σ ∈ [0,∞]. For every n ≥ 1, we call qn a random map uniformly distributed over the set of all rooted planar quadrangulations with a boundary having n faces and 2σn half-edges on the boundary. For σ ∈ (0,∞), we view qn as a metric space by endowing its set of vertices with the graph metric, rescaled by n. 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 −1/2 n and we obtain a convergence toward Aldous’s CRT.