[Paper Review] Convergence of odd-angulations via symmetrization of labeled trees

Louigi Addario‐Berry, Marie Albenque|arXiv (Cornell University)|Apr 9, 2019

Stochastic processes and statistical mechanics21 references9 citations

TL;DR

This paper establishes the convergence of uniform random $p$-angulations with $p \geq 5$ odd to the Brownian map under Gromov-Hausdorff-Prokhorov topology after rescaling distances by $C_p/n^{1/4}$, using a novel symmetrization technique for labeled multitype trees and an invariance principle for critical Boltzmann trees with weak centering assumptions on label displacements.

ABSTRACT

Fix $p\geq 5$ an odd integer integer. Let $M_n$ be a uniform $p$-angulation with $n$ vertices and endowed with the uniform probability measure on its vertices. We prove that, there exists $C_p\in \mathbb{R}_+$ such that, after rescaling distances by $C_p/n^{1/4}$, $M_n$ converges in distribution for the Gromov-Hausdorff-Prokhorov topology towards the Brownian map. To prove the preceding fact, we introduce a `bootstrapping' principle for distributional convergence of random labelled plane trees. In particular, the latter allows to obtain an invariance principle for labeled multitype Galton-Watson trees, with only a weak assumption on the centering of label displacements

Motivation & Objective

• To establish the universality of the Brownian map as the scaling limit for non-bipartite random planar maps, specifically for $p$-angulations with odd $p \geq 5$.
• To overcome the lack of bipartiteness in odd-angulations, which prevents direct application of existing convergence results relying on bipartite structure.
• To develop a new bootstrapping principle for distributional convergence of labeled plane trees, applicable under weak moment conditions on label displacements.
• To extend the convergence framework to multitype labeled trees by introducing symmetrization to handle non-centered label increments.
• To prove that the encoding of $p$-angulations via the Bouttier-di Francesco-Guitter bijection leads to a process converging to the Brownian snake, thereby implying convergence to the Brownian map.

Proposed method

• Introduce a symmetrization procedure for labeled multitype trees to handle non-centered label displacements, enabling convergence results without requiring centered increments.
• Develop a bootstrapping principle that allows distributional convergence of labeled tree processes to the Brownian snake under minimal assumptions on label variance and tree type structure.
• Apply the symmetrization technique to critical Boltzmann maps with $p$-angulations, showing that their tree encodings converge to the Brownian snake in distribution.
• Use the Bouttier-di Francesco-Guitter bijection to map $p$-angulations to labeled multitype trees, preserving the geometric structure needed for scaling limits.
• Leverage the convergence of tree encoding processes to the Brownian snake, combined with known results from Le Gall and Miermont, to deduce convergence of the original maps to the Brownian map.
• Employ Skorohod representation and almost sure convergence arguments to eliminate subsequence extraction, proving that the limiting distance process is exactly the Brownian map's distance function.

Experimental results

Research questions

• RQ1 Can the Brownian map be the universal scaling limit for non-bipartite random planar maps, including odd-angulations?
• RQ2 How can convergence results for labeled trees be extended to multitype trees with non-centered label displacements?
• RQ3 What symmetrization technique enables distributional convergence of labeled trees without requiring centered label increments?
• RQ4 Does the encoding of $p$-angulations via labeled trees converge to the Brownian snake for odd $p \geq 5$?
• RQ5 Can the convergence of the tree encoding be used to deduce convergence of the original maps to the Brownian map in the Gromov-Hausdorff-Prokhorov topology?

Key findings

• For any odd integer $p \geq 5$, uniform $p$-angulations with $n$ vertices converge in distribution to the Brownian map under Gromov-Hausdorff-Prokhorov topology after rescaling distances by $C_p / n^{1/4}$.
• The constant $C_p$ depends only on $p$ and ensures the correct scaling for convergence to the Brownian map.
• A new bootstrapping principle for labeled trees is developed, allowing convergence under weak moment assumptions on label displacements, without requiring centered increments.
• The symmetrization of labeled trees enables the extension of invariance principles to multitype Galton-Watson trees with only minimal centering assumptions.
• The convergence of the tree encoding process to the Brownian snake is established, which implies the convergence of the original maps to the Brownian map via known bijection results.
• The limiting distance process is shown to be exactly the Brownian map's distance function $D^*$, by proving almost sure equality of the limiting process with $D^*$ using uniform vertex sampling and convergence in distribution.