Probability Seminar: Delphin Sénizergues

Starting from a sequence of positive real numbers (w_n), which we call weights, we construct a tree in a recursive manner: at time 1, the tree has only one vertex. Then at any step n+1, we add a new vertex to the tree and we choose its parent at random among the already existing vertices, in such a way that the k-th vertex (in order of creation) is chosen with probability proportional to w_k.

This model generalises the well-known uniform recursive tree (URT) in the case of a constant sequence (w_n). In fact, it can also be shown hat the trees constructed using affine preferential attachment can be described with this construction, using a random sequence of weights (w_n).

We prove almost-sure scaling limits for the height, profile and degrees in the tree as the number of vertices tends to infinity. These results are related to proving scaling limits in the Gromov-Hausdorff-Prokhorov topology for a family of random growth models on graphs that generalises Rémy's algorithm.