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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 $D_t(\ell)$ of vertices of each degree $\ell$ at each time $t$, focussing particularly on the case where $\ell$ is a growing function of $t$. We show that $D_t(\ell)$ is concentrated around its mean, which is approximately $4t/\ell^3$, for all $\ell \le (t/\log t)^{-1/3}$; this is best possible up to a logarithmic factor.