The distribution of the maximum vertex degree in random planar maps

We determine the limiting distribution of the maximum vertex degree Delta(n) 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 G(2), where G(2) 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 Delta(n). 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. (C) 2000 Academic Press.