25 research outputs found
The continuum random tree is the scaling limit of unlabelled unrooted trees
We prove that the uniform unlabelled unrooted tree with n vertices and vertex
degrees in a fixed set converges in the Gromov-Hausdorff sense after a suitable
rescaling to the Brownian continuum random tree. This proves a conjecture by
Aldous. Moreover, we establish Benjamini-Schramm convergence of this model of
random trees
Random enriched trees with applications to random graphs
We establish limit theorems that describe the asymptotic local and global
geometric behaviour of random enriched trees considered up to symmetry. We
apply these general results to random unlabelled weighted rooted graphs and
uniform random unlabelled -trees that are rooted at a -clique of
distinguishable vertices. For both models we establish a Gromov--Hausdorff
scaling limit, a Benjamini--Schramm limit, and a local weak limit that
describes the asymptotic shape near the fixed root