[Paper Review] The boundary of random planar maps via looptrees

Igor Kortchemski, Loïc Richier|arXiv (Cornell University)|Feb 2, 2018

Stochastic processes and statistical mechanics4 citations

TL;DR

This paper establishes the scaling limits of looptrees derived from conditioned Bienaymé–Galton–Watson trees under different offspring distributions. For critical, finite-variance offspring distributions, the scaling limit is the Brownian continuum random tree; for heavy-tailed, subcritical offspring distributions, the limit is a multiple of the unit circle, reflecting a condensation phenomenon. These results resolve a prediction from [CK14] and complete the phase transition analysis of large faces in Boltzmann planar maps in the non-generic critical regime.

ABSTRACT

We study the scaling limits of looptrees associated with Bienaymé--Galton--Watson (BGW) trees, that are obtained by replacing every vertex of the tree by a "cycle" whose size is its degree. First, we consider BGW trees whose offspring distribution is critical and in the domain of attraction of a Gaussian distribution. We prove that the Brownian CRT is the scaling limit of the associated looptrees, thereby confirming a prediction of [CK14b]. Then, we deal with BGW trees whose offspring distribution is critical and heavy-tailed. We show that the scaling limit of the associated looptrees is a multiple of the unit circle. This corresponds to a so-called condensation phenomenon, meaning that the underlying tree exhibits a vertex with macroscopic degree. Here, we rely on an invariance principle for random walks with negative drift, which is of independent interest. Finally, we apply these results to the study of the scaling limits of large faces of Boltzmann planar maps. We complete the results of [Ric17] and establish a phase transition for the topology of these maps in the non-generic critical regime.

Motivation & Objective

• To establish the scaling limit of looptrees associated with critical, finite-variance Bienaymé–Galton–Watson trees, confirming a prediction from [CK14].
• To analyze the scaling limit of looptrees under heavy-tailed, subcritical offspring distributions, where a condensation phenomenon occurs due to a macroscopic-degree vertex.
• To apply these looptree limits to characterize the topology of large faces in Boltzmann planar maps, particularly in the non-generic critical regime.
• To develop an invariance principle for random walks with negative drift, which is instrumental in the heavy-tailed case.

Proposed method

• Construct looptrees by replacing each vertex in a plane tree with a cycle of length equal to its degree, preserving the tree's combinatorial structure.
• Use the Gromov–Hausdorff topology to study the scaling limits of rescaled looptrees as the number of vertices tends to infinity.
• For the critical, finite-variance case, apply a functional central limit theorem to the exploration process of the tree, leading to convergence to the Brownian CRT.
• For the heavy-tailed, subcritical case, employ a spinal decomposition and an invariance principle for random walks with negative drift to show convergence to a multiple of the unit circle.
• Leverage the contour function of the tree and its relation to the looptree's metric structure to derive scaling limits.
• Apply the looptree results to Boltzmann planar maps by analyzing the scaling limit of their boundary components, using the looptree as a model for large faces.

Experimental results

Research questions

• RQ1 What is the scaling limit of looptrees associated with critical, finite-variance Bienaymé–Galton–Watson trees?
• RQ2 How do looptrees associated with subcritical, heavy-tailed offspring distributions scale, and what role does condensation play?
• RQ3 Can the invariance principle for random walks with negative drift be used to derive scaling limits of looptrees in the heavy-tailed regime?
• RQ4 What is the topological phase transition in the boundary of large Boltzmann planar maps in the non-generic critical regime?
• RQ5 How do the results on looptrees complete the understanding of the scaling limits of large faces in random planar maps?

Key findings

• For critical, finite-variance offspring distributions, the rescaled looptrees converge in distribution to the Brownian continuum random tree under the Gromov–Hausdorff topology.
• For subcritical, heavy-tailed offspring distributions with tail index β > 1, the rescaled looptrees converge to a multiple of the unit circle, indicating a condensation phenomenon due to a vertex of macroscopic degree.
• The convergence in the heavy-tailed case relies on a novel invariance principle for random walks with negative drift, which is of independent interest.
• The results confirm a prediction from [CK14] regarding the scaling limit of looptrees in the critical, finite-variance case.
• The paper completes the phase transition analysis for the topology of large faces in Boltzmann planar maps in the non-generic critical regime, showing a transition from tree-like to circular boundary structures.