Invariance principles for random bipartite planar maps

Abstract

Random planar maps are considered in the physics literature as the discrete counterpart of random surfaces. It is conjectured that properly rescaled random planar maps, when conditioned to have a large number of faces, should converge to a limiting surface whose law does not depend, up to scaling factors, on details of the class of maps that are sampled. Previous works on the topic, starting with Chassaing and Schaeffer, have shown that the radius of a random quadrangulation with $n$ faces, that is, the maximal graph distance on such a quadrangulation to a fixed reference point, converges in distribution once rescaled by $n^{1/4}$ to the diameter of the Brownian snake, up to a scaling constant. Using a bijection due to Bouttier, Di Francesco and Guitter between bipartite planar maps and a family of labeled trees, we show the corresponding invariance principle for a class of random maps that follow a Boltzmann distribution putting weight $q_k$ on faces of degree $2k$: the radius of such maps, conditioned to have $n$ faces (or $n$ vertices) and under a criticality assumption, converges in distribution once rescaled by $n^{1/4}$ to a scaled version of the diameter of the Brownian snake. Convergence results for the so-called profile of maps are also provided. The convergence of rescaled bipartite maps to the Brownian map, in the sense introduced by Marckert and Mokkadem, is also shown. The proofs of these results rely on a new invariance principle for two-type spatial Galton--Watson trees.

Published in at http://dx.doi.org/10.1214/009117906000000908 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (3)

• Uniform Infinite Planar Triangulations
• Local limit of labeled trees and expected volume growth in a random quadrangulation
• Limit of normalized quadrangulations: The Brownian map

Cited by in corpus (18)

• Uniqueness and universality of the Brownian map
• The scaling limit of uniform random plane maps, via the Ambjørn-Budd bijection
• Scaling limits of random planar maps with a unique large face
• Recurrence of bipartite planar maps
• The two uniform infinite quadrangulations of the plane have the same law
• Scaling limits of random bipartite planar maps with a prescribed degree sequence
• A Boltzmann approach to percolation on random triangulations
• Spanning forests in regular planar maps
• On scaling limits of planar maps with stable face-degrees
• On scaling limits of random trees and maps with a prescribed degree sequence
• The lineage process in Galton--Watson trees and globally centered discrete snakes
• On the diameter of random planar graphs
• Scaling limits of discrete snakes with stable branching
• Large deviation Local Limit Theorems and limits of biconditioned Trees and Maps
• Limit Laws of Planar Maps with Prescribed Vertex Degrees
• The vertical profile of embedded trees
• Scaling limits of random looptrees and bipartite plane maps with prescribed large faces
• Random Lévy Looptrees and Lévy Maps