The Distribution of the Maximum Vertex Degree in Random Planar Maps

We determine the limiting distribution of the maximum vertex degree Δn in a random triangulation of an n-gon, and show that it is the same as that of the maximum of n independent identically distributed random variables G2, where G2 is the sum of two independent geometric(1/2) random variables. This answers affirmatively a question of Devroye, Flajolet, Hurtado, Noy and Steiger, who gave much weaker almost sure bounds on Δn. An interesting consequence of this is that the asymptotic probability that a random triangulation has a unique vertex with maximum degree is about 0.72. We also give an analogous result for random planar maps in general.