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

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.

46 pages, 5 figures. Minor revisions since the first version

References in corpus (4)

• The topological structure of scaling limits of large planar maps
• The exploration process of inhomogeneous continuum random trees, and an extension of Jeulin's local time identity
• A Boltzmann approach to percolation on random triangulations
• The lineage process in Galton--Watson trees and globally centered discrete snakes

Cited by in corpus (6)

• Local limits of uniform triangulations in high genus
• On scaling limits of random trees and maps with a prescribed degree sequence
• Scaling limits of discrete snakes with stable branching
• Large deviation Local Limit Theorems and limits of biconditioned Trees and Maps
• Scaling limits of random looptrees and bipartite plane maps with prescribed large faces
• Unified study of the phase transition for block-weighted random planar maps