Open Access
Translator Disclaimer
January 2011 Scaling limits of random planar maps with large faces
Jean-François Le Gall, Grégory Miermont
Ann. Probab. 39(1): 1-69 (January 2011). DOI: 10.1214/10-AOP549

Abstract

We discuss asymptotics for large random planar maps under the assumption that the distribution of the degree of a typical face is in the domain of attraction of a stable distribution with index α∈(1, 2). When the number n of vertices of the map tends to infinity, the asymptotic behavior of distances from a distinguished vertex is described by a random process called the continuous distance process, which can be constructed from a centered stable process with no negative jumps and index α. In particular, the profile of distances in the map, rescaled by the factor n−1∕2α, converges to a random measure defined in terms of the distance process. With the same rescaling of distances, the vertex set viewed as a metric space converges in distribution as n→∞, at least along suitable subsequences, toward a limiting random compact metric space whose Hausdorff dimension is equal to 2α.

Citation

Download Citation

Jean-François Le Gall. Grégory Miermont. "Scaling limits of random planar maps with large faces." Ann. Probab. 39 (1) 1 - 69, January 2011. https://doi.org/10.1214/10-AOP549

Information

Published: January 2011
First available in Project Euclid: 3 December 2010

zbMATH: 1204.05088
MathSciNet: MR2778796
Digital Object Identifier: 10.1214/10-AOP549

Subjects:
Primary: 05C80 , 60F17 , 60G51

Keywords: graph distance , Gromov–Hausdorff convergence , Hausdorff dimension , profile of distances , Random planar map , Scaling limit , stable distribution , stable tree

Rights: Copyright © 2011 Institute of Mathematical Statistics

JOURNAL ARTICLE
69 PAGES


SHARE
Vol.39 • No. 1 • January 2011
Back to Top