Lifting linear preferential attachment trees yields the arcsine coalescent

TL;DR

This paper demonstrates that repeatedly lifting linear preferential attachment trees produces a partition process equivalent to the arcsine $n$-coalescent, linking tree growth dynamics to a well-known coalescent process.

Contribution

It establishes a novel connection between lifting operations on preferential attachment trees and the arcsine coalescent process, providing new insights into their probabilistic structure.

Findings

The partition process from lifted trees matches the arcsine $n$-coalescent in distribution.

Lifting operations induce a Markov chain on preferential attachment trees.

The resulting coalescent process involves multiple mergers governed by the arcsine distribution.

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.