64 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
Graph limits of random graphs from a subset of connected -trees
For any set of non-negative integers such that and , we consider a random --tree that is uniformly selected from all connected -trees of
vertices where the number of -cliques that contain any fixed -clique
belongs to . We prove that , scaled by
where is the -th Harmonic
number and , converges to the Continuum Random Tree
. Furthermore, we prove the local convergence of the
rooted random --tree to an infinite but
locally finite random --tree .Comment: 21 pages, 6 figure
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