The Brownian Plane with Minimal Neck Baby Universe

For each $n\in\mathbb{N}$, let $\mathbf{Q}_n$ be a uniform rooted measured quadrangulation of size $n$ conditioned to have $r(n)$ vertices in its root block. We prove that for a suitable function $r(n)$, after rescaling graph distance by $\left(\frac{21}{40\cdot r(n)}\right)^{1/4}$, with an appropriate rescaling of measure, $\mathbf{Q}_n$ converges to a random pointed measured non-compact metric space $\boldsymbol{ \mathcal{S}}$, in the local Gromov-Hausdorff-Prokhorov topology; the space $\boldsymbol{ \mathcal{S}}$ is built by identifying a uniform point of the Brownian map with the distinguished point of the Brownian plane. Our result relies upon both the convergence of uniform quadrangulations towards the Brownian plane by \citet{CLG}, and the convergence of uniform $2$-connected quadrangulations to the Brownian map, recently proved by \citet{ABW}. The main steps of the proof are as follows. First, we show that the sizes of submaps pendant to the root block have an asymptotically stable distribution. Second, we deduce asymptotics for occupancy in a random allocation model with a varying balls-to-boxes ratio. Third, we establish a bound for the number of pendant submaps of the root block, which allow us to apply the occupancy bounds to uniformly control the sizes of pendant submaps. This entails us to show that the pendant submaps act as uniformly asymptotically negligible ``decorations'' which do not affect the scaling limit.