Vertices of high degree in the preferential attachment tree

We study the basic preferential attachment process, which generates a sequence of random trees, each obtained from the previous one by introducing a new vertex and joining it to one existing vertex, chosen with probability proportional to its degree. We investigate the number D t(l) of vertices of each degree l at each time t, focussing particularly on the case where l is a growing function of t. We show that D t(l) is concentrated around its mean, which is approximately 4t=l 3, for all l ≤ (t= log t) -1/3; this is best possible up to a logarithmic factor.