Probability Seminar: Abbas Mehrabian

Consider the following iterative construction of a random planar triangulation. Start with a triangle embedded in the plane. In each step, choose a bounded face uniformly at random, add a vertex inside that face and join it to the vertices of the face. After n – 3 steps, we obtain a random triangulated plane graph with n vertices, which is called a Random Apollonian Network (RAN). See http://www.math.cmu.edu/~ctsourak/ran.html for an example. We prove that the diameter of a RAN is asymptotic to c log(n) in probability, where c ≈ 1.668 is the solution of an explicit equation. The proof adapts a technique of Broutin and Devroye for estimating the height of random trees. We also prove that there exists a fixed s1, such that eventually every self-avoiding walk in this graph has length less than n^s, which verifies a conjecture of Cooper and Frieze. Using a similar technique, we show that if r d are fixed constants, then every r-ary subtree of a random d-ary recursive tree on n vertices has less than n^b vertices, for some b=b(d,r)1. Based on joint work with A. Collevecchio, E. Ebrahimzadeh, L. Farczadi, P. Gao, C. Sato, N. Wormald, and J. Zung.