Scaling limits and influence of the seed graph in preferential attachment trees

We are interested in the asymptotics of random trees built by linear preferential attachment, also known in the literature as Barabási–Albert trees or plane-oriented recursive trees. We first prove a conjecture of Bubeck, Mossel & Rácz [7] concerning the influence of the seed graph on the asymptotic behavior of such trees. Separately we study the geometric structure of nodes of large degrees in a plane version of Barabási–Albert trees via their associated looptrees. As the number of nodes grows, we show that these looptrees, appropriately rescaled, converge in the Gromov– Hausdorff sense towards a random compact metric space which we call the Brownian looptree. The latter is constructed as a quotient space of Aldous’ Brownian Continuum Random Tree and is shown to have almost sure Hausdorff dimension 2. Figure 1: The looptree associated with a large plane Barabási–Albert tree. ♠CNRS and LPMA, Université Pierre et Marie Curie (Paris 6). [email protected] ♥LPMA, Université Pierre et Marie Curie (Paris 6). [email protected] ♦DMA, École Normale Supérieure. [email protected] ♣Département de Mathématiques, Université de Genève. [email protected] MSC2010 subject classifications. Primary 05C80, 60J80; secondary 05C05, 60G42.