Critical And Subcritical Scaling Limits Of Random Planar Maps With Connectivity Constraints PDF Download

"In the first part, we show that a uniform quadrangulation, its largest 2-connected block, and its largest simple block, upon rescaling the graph distance properly, jointly converge to the same Brownian map in distribution for the Gromov-Hausdorff-Prokhorov topology. We start by deriving a local limit theorem for the asymptotics of maximal block sizes, extending the result by Banderier, Flajolet, Schaeffer & Soria[16]. The resulting diameter bounds for pendant submaps of random quadrangulations straightforwardly lead to Gromov-Hausdorff convergence. To extend the convergence to the Gromov-Hausdorff-Prokhorov topology, we show that exchangeable "uniformly asymptotically negligible" attachments of mass simply yield, in the limit, a deterministic scaling of the mass measure. In the second part, for each n ∈ N, let Qn 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, Qn converges to a random pointed measured non-compact metric space S, in the local Gromov-Hausdorff-Prokhorov topology; the space 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 Curien & Le Gall[30], and the convergence of uniform 2-connected quadrangulations to the Brownian map, proved in the first part of the thesis. 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 allows us to apply the occupancy bounds to uniformly control the sizes of pendant submaps. This entails that the pendant submaps act as uniformly asymptotically negligible "decorations" which do not affect the scaling limit." --