On scaling limits of random trees and maps with a prescribed degree sequence

TL;DR

This paper investigates the scaling limits of random bipartite planar maps with prescribed face degrees, showing convergence to Brownian objects under certain conditions and connecting to stable laws.

Contribution

It establishes the convergence of these maps to Brownian spheres, disks, or trees depending on face degree distributions, extending known results and providing new geometric insights.

Findings

Convergence to Brownian sphere and disk under specific conditions

Maps degenerate to Brownian tree when large faces dominate

Results connect face degree distributions to stable laws and geometric limits

Abstract

We study a configuration model on bipartite planar maps in which, given $n$ even integers, one samples a planar map with $n$ faces uniformly at random with these face degrees. We prove that when suitably rescaled, such maps always admit nontrivial subsequential limits as $n \to \infty$ in the Gromov-Hausdorff-Prokhorov topology. Further, we show that they converge in distribution towards the celebrated Brownian sphere, and more generally a Brownian disk for maps with a boundary, if and only if there is no inner face with a macroscopic degree, or, if the perimeter is too big, the maps degenerate and converge to the Brownian tree. By first sampling the degrees at random with an appropriate distribution, this model recovers that of size-conditioned Boltzmann maps associated with critical weights in the domain of attraction of a stable law with index $\alpha\in [1,2]$. The Brownian tree and disks then appear respectively in the case $\alpha=1$ and $\alpha=2$, whereas in the case $\alpha \in (1,2)$ our results partially recover previous known ones. Our proofs rely on known bijections with labelled plane trees, which are similarly sampled uniformly at random given $n$ outdegrees. Along the way, we obtain some results on the geometry of such trees, such as a convergence to the Brownian tree but only in the weaker sense of subtrees spanned by random vertices, which are of independent interest.