On the growth of random planar maps with a prescribed degree sequence

TL;DR

This paper studies the asymptotic geometric properties of random bipartite planar maps with prescribed face degrees, showing their diameter scales with a global variance and establishing subsequential Gromov-Hausdorff-Prokhorov limits.

Contribution

It introduces a novel approach using a bijection with labeled trees and a new spinal decomposition to analyze the maps' scaling limits, even when the underlying trees are not tight.

Findings

Map diameters grow like the square root of the variance term

Vertex sets with scaled graph distance have subsequential Gromov-Hausdorff-Prokhorov limits

Label processes are tight under suitable rescaling despite non-tight underlying trees

Abstract

For non-negative integers $(d_n(k))_{k \ge 1}$ such that $\sum_{k \ge 1} d_n(k) = n$, we sample a bipartite planar map with $n$ faces uniformly at random amongst those which have $d_n(k)$ faces of degree $2k$ for every $k \ge 1$ and we study its asymptotic behaviour as $n \to \infty$. We prove that the diameter of such maps grow like $\sigma_n^{1/2}$, where $\sigma_n^2 = \sum_{k \ge 1} k (k-1) d_n(k)$ is a global variance term. More precisely, we prove that the vertex-set of these maps equipped with the graph distance divided by $\sigma_n^{1/2}$ and the uniform probability measure always admits subsequential limits in the Gromov-Hausdorff-Prokhorov topology. Our proof relies on a bijection with random labelled trees; we are able to prove that the label process is always tight when suitably rescaled, even if the underlying tree is not tight for the Gromov-Hausdorff topology. We also rely on a new spinal decomposition which is of independent interest. Finally this paper also serves as a toolbox for a companion paper in which we discuss more precisely Brownian limits of such maps.