Geodesic
Boundaries of a random triangulation of a disk
Diskretnaya Matematika, Tome 16 (2004) no. 2, pp. 121-135.

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We consider random triangulations of a disk with $k$ holes and $N$ triangles as $N\to\infty$. The coefficient $\lambda^m$, $\lambda>0$, is assigned to a triangulation with the total number of boundary edges equal to $m$. In the case of two boundaries, we separate three domains of variation of the parameter $\lambda$, and in each of them find the limit joint distribution of boundary lengths. For a greater number of boundaries, we give an algorithm to calculate the generating functions for the number of multi-rooted triangulations depending of the number of triangles and the lengths of boundaries. In Appendix, we discuss the relation between multi-rooted triangulations and unrooted triangulations, and give analogues of limit distributions for unrooted triangulations. This research was supported by the Russian Foundation for Basic Research, grant 02–01–00415. 
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M. A. Krikun. Boundaries of a random triangulation of a disk. Diskretnaya Matematika, Tome 16 (2004) no. 2, pp. 121-135. http://geodesic.mathdoc.fr/item/DM_2004_16_2_a9/

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