Phase transition for tree-rooted maps

TL;DR

This paper introduces a weighted model of tree-rooted planar maps, analyzes its phase transition, and characterizes the size distribution of 2-connected blocks, showing convergence to the Brownian Continuum Random Tree in critical regimes.

Contribution

It presents a new weighted model for tree-rooted maps, proves the existence of a phase transition, and describes the scaling limits in different regimes.

Findings

Phase transition identified in the model.

Distribution of largest 2-connected blocks characterized.

Scaling limits converge to Brownian CRT in critical regimes.

Abstract

We introduce a model of tree-rooted planar maps weighted by their number of $2$-connected blocks. We study its enumerative properties and prove that it undergoes a phase transition. We give the distribution of the size of the largest $2$-connected blocks in the three regimes (subcritical, critical and supercritical) and further establish that the scaling limit is the Brownian Continuum Random Tree in the critical and supercritical regimes, with respective rescalings $\sqrt{n/\log(n)}$ and $\sqrt{n}$.