The topological structure of scaling limits of large planar maps. (English) Zbl 1132.60013
The purpose of this work is to investigate continuous limits of rescaled planar maps. Let \(p\) be a fixed integer \((p\geq2)\) and consider for every integer \(n\geq2\) a random planar map \(M_n\) which is uniformly distributed over the set of all rooted \(2p\)-angulations with \(n\) faces. Then, at least along a suitable subsequence, the metric space consisting of the set of vertices of \(M_n\), equipped with the graph distance rescaled by the factor \(n^{-1/4}\), converges in distribution as \(n\to\infty\) towards a limiting random compact metric space, in the sense of the Gromov-Hausdorff distance. The author proves that the topology of the limiting space is uniquely determined independently of \(p\) and of the subsequence, and that this space can be obtained as the quotient of the continuum random tree for an equivalence relation which is defined from Brownian labels attached to the vertices.
Finally, the last section contains the calculation of the Hausdorff dimension of the limiting metric space.
Finally, the last section contains the calculation of the Hausdorff dimension of the limiting metric space.
Reviewer: Nicko G. Gamkrelidze (Moskva)
MSC:
| 60E05 | Probability distributions: general theory |
| 60B05 | Probability measures on topological spaces |
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