Scaling limits of random bipartite planar maps with a prescribed degree sequence

TL;DR

This paper investigates the asymptotic geometric behavior of large bipartite planar maps with fixed face-degree sequences, demonstrating convergence to the Brownian map under mild conditions, extending previous results and applying to various distributions.

Contribution

It establishes the convergence of bipartite maps with prescribed degrees to the Brownian map, generalizing prior work and including Boltzmann-distributed maps under minimal assumptions.

Findings

Convergence of rescaled bipartite maps to the Brownian map in Gromov-Hausdorff sense

Extension of previous results to maps with prescribed face-degree sequences

Applicability to Boltzmann-distributed maps under second moment condition

Abstract

We study the asymptotic behaviour of uniform random maps with a prescribed face-degree sequence, in the bipartite case, as the number of faces tends to infinity. Under mild assumptions, we show that, properly rescaled, such maps converge in distribution towards the Brownian map in the Gromov-Hausdorff sense. This result encompasses a previous one of Le Gall for uniform random $q$-angulations where $q$ is an even integer. It applies also to random maps sampled from a Boltzmann distribution, under a second moment assumption only, conditioned to be large in either of the sense of the number of edges, vertices, or faces. The proof relies on the convergence of so-called "discrete snakes" obtained by adding spatial positions to the nodes of uniform random plane trees with a prescribed child sequence recently studied by Broutin & Marckert. This paper can alternatively be seen as a contribution to the study of the geometry of such trees.