The Annals of Probability

Scaling limits of random planar maps with large faces

Jean-François Le Gall and Grégory Miermont

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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.

Article information

Source
Ann. Probab., Volume 39, Number 1 (2011), 1-69.

Dates
First available in Project Euclid: 3 December 2010

Permanent link to this document
https://projecteuclid.org/euclid.aop/1291388296

Digital Object Identifier
doi:10.1214/10-AOP549

Mathematical Reviews number (MathSciNet)
MR2778796

Zentralblatt MATH identifier
1204.05088

Subjects
Primary: 05C80: Random graphs [See also 60B20] 60F17: Functional limit theorems; invariance principles 60G51: Processes with independent increments; Lévy processes

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

Citation

Le Gall, Jean-François; Miermont, Grégory. Scaling limits of random planar maps with large faces. Ann. Probab. 39 (2011), no. 1, 1--69. doi:10.1214/10-AOP549. https://projecteuclid.org/euclid.aop/1291388296


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