Lifting linear preferential attachment trees yields the arcsine coalescent

Abstract

We consider linear preferential attachment trees which are specific scale-free trees also known as (random) plane-oriented recursive trees. Starting with a linear preferential attachment tree of size $n$ we show that repeatedly applying a so-called lifting yields a continuous-time Markov chain on linear preferential attachment trees. Each such tree induces a partition of $\{1, \ldots, n\}$ by placing labels in the same block if and only if they are attached to the same node in the tree. Our main result is that this Markov chain on linear preferential attachment trees induces a partition valued process which is equal in distribution (up to a random time-change) to the arcsine $n$-coalescent, that is the multiple merger coalescent whose $\Lambda$ measure is the arcsine distribution.