Electronic Journal of Probability

Asymptotics for Rooted Bipartite Planar Maps and Scaling Limits of Two-Type Spatial Trees

Mathilde Weill

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We prove some asymptotic results for the radius and the profile of large random bipartite planar maps. Using a bijection due to Bouttier, Di Francesco and Guitter between rooted bipartite planar maps and certain two-type trees with positive labels, we derive our results from a conditional limit theorem for two-type spatial trees. Finally we apply our estimates to separating vertices of bipartite planar maps: with probability close to one when n tends to infinity, a random $2k$-angulation with n faces has a separating vertex whose removal disconnects the map into two components each with size greater that $n^{1/2 - \varepsilon}$.

Article information

Electron. J. Probab., Volume 12 (2007), paper no. 31, 862-925.

Accepted: 13 June 2007
First available in Project Euclid: 1 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 05C80: Random graphs [See also 60B20]
Secondary: 60F17: Functional limit theorems; invariance principles 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

Planar maps two-type Galton-Watson trees Conditioned Brownian snake

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Weill, Mathilde. Asymptotics for Rooted Bipartite Planar Maps and Scaling Limits of Two-Type Spatial Trees. Electron. J. Probab. 12 (2007), paper no. 31, 862--925. doi:10.1214/EJP.v12-425. https://projecteuclid.org/euclid.ejp/1464818502

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