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.
Publié le : 2011-01-15
Classification:
Random planar map,
scaling limit,
graph distance,
profile of distances,
stable distribution,
stable tree,
Gromov–Hausdorff convergence,
Hausdorff dimension,
05C80,
60F17,
60G51
@article{1291388296,
author = {Le Gall, Jean-Fran\c cois and Miermont, Gr\'egory},
title = {Scaling limits of random planar maps with large faces},
journal = {Ann. Probab.},
volume = {39},
number = {1},
year = {2011},
pages = { 1-69},
language = {en},
url = {http://dml.mathdoc.fr/item/1291388296}
}
Le Gall, Jean-François; Miermont, Grégory. Scaling limits of random planar maps with large faces. Ann. Probab., Tome 39 (2011) no. 1, pp. 1-69. http://gdmltest.u-ga.fr/item/1291388296/