Boundaries of a random triangulation of a disk

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.