Joint convergence of random quadrangulations and their cores

Abstract

We show that a uniform quadrangulation, its largest 2-connected block, and its largest simple block 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 in \cite{BFSS}. 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.

30 pages, 5 figures. Minor changes for v2

References in corpus (2)

• Uniqueness and universality of the Brownian map
• The scaling limit of random simple triangulations and random simple quadrangulations

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