Scaling Limit of Random Planar Quadrangulations with a Boundary

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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⁻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 σₙ⁻¹/² and we obtain a convergence toward Aldous's CRT.