SCALING LIMITS OF RANDOM PLANAR MAPS WITH LARGE FACES

• 成果类型: Article
• 署名作者: Le Gall, Jean-Francois; Miermont, Gregory
• 署名单位: Universite Paris Saclay
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/10-AOP549
• 发表日期: 2011
• 页码: 1-69
• 关键词: trees

摘要

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 alpha is an element of (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 alpha. In particular, the profile of distances in the map, rescaled by the factor n(-1/2 alpha), 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 -> infinity, at least along suitable subsequences, toward a limiting random compact metric space whose Hausdorff dimension is equal to 2 alpha.