A Tauberian approach to metric scaling limits of random discrete structures, with an application to random planar maps

TL;DR

The paper develops a Tauberian framework in the GHP space using a Lipschitz-type order to deduce metric scaling limits from component models to host models. It proves sandwich and Tauberian theorems in this setting and applies them to random planar maps, showing that irreducible quadrangulations converge to the Brownian sphere under GHP with appropriate normalizations. The method hinges on condensation to identify a large irreducible core, a topological/measure-theoretic transfer of convergence from core to whole maps, and a Tauberian step that passes from random to deterministic sizes via monotone couplings (face-openings) for an irreducible hexagon. The approach thus yields a robust, modular pathway to derive scaling limits in random maps, with potential applicability to other combinatorial models and their metric scaling limits.

Abstract

We prove sandwich theorems and a Tauberian theorem in the space of compact metric measure spaces, endowed with the Gromov-Hausdorff-Prokhorov (GHP) topology. These results hold with respect to a close relative of Gromov's Lipschitz order. As a proof-of-concept of a general method to prove metric scaling limits of random discrete structures, we give an application to the theory of random planar maps: the Brownian sphere is the scaling limit in the GHP topology of irreducible quandrangulations. Our main inputs are (i) the convergence of general quadrangulations to the Brownian sphere (Le Gall, 2013; Miermont, 2013); and (ii) couplings where irreducible quadrangulations of the hexagon are "grown" by face-openings (Addario-Berry, 2014).

Figures (7)

• Figure 1: A schematic representation of the setup of Lemma \ref{['lem:informal-lemma-star-decomp']}.
• Figure 2: Setup of the proof of Lemma \ref{['lem:intersection-neighborhood']}. The geodesic path $p\colon [0,\delta(x,y)]\to X\cup Y$ such that $p(0)=x$ and $p(\delta(x,y))=y$ has been represented in red.
• Figure 3: The "comb" counter-example for Remark \ref{['rem:comb-counter-example']}. Fix any continuous function $f\colon[0,1]\to {\mathbb R}_+$. We let $Z_n\subset{\mathbb R}^2$ be defined by $Z_n=X_n\cup\bigsqcup_i Y^{(i)}_n$, where $X_n=[0,1]\times\{0\}$ is equipped with its Lebesgue measure, and for every $n$, the segment $Y^{(i)}_n=\{i/n\}\times[0,\frac{1}{n} f(i/n)]$ is also equipped with its Lebesgue measure. Then we replace the induced metric on $Z_n\subset{\mathbb R}^2$ by its length metric$d_n$ induced by the length of shortest paths inside$Z_n$, so as to turn $Z_n$ into a geodesic space. It is easily seen that $(Z_n,d_n,\mu_n)$ converges in the GHP metric to $[0,1]$ with its usual distance, and equipped with the measure $(1+f(x))\mathrm{Leb}(\mathop{}\!\mathrm{d} x)$. Such a convergence can be made to hold in the Hausdorff+Prokhorov sense inside some compact space $(Z,\delta)$ by Proposition \ref{['prop:GHP-common-embedding']}
• Figure 4: Depiction of how $\mathrm{Open}(\mathfrak{q},e,e')$ is constructed, starting from a quadrangulation $\mathfrak{q}$ and two edges $e,e'$ sharing one endpoint.
• Figure 5: The "pattern" ${\mathtt P_0}$, which is a specific irreducible quadrangulation of a hexagon which has 9 interior faces.

Theorems & Definitions (107)

• Definition 1.1
• Theorem 1.2: GHP Sandwich theorem
• Theorem 1.3: Probabilistic GHP Sandwich theorem
• Theorem 1.4: GHP Tauberian theorem
• Remark 1.5
• Theorem 1.6: Le Gall, 2013; Miermont, 2013
• Theorem 1.7
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3