Scaling limit of random k-trees

Abstract

We consider a random k-tree G n,k that is uniformly selected from the class of labelled k-trees with n + k vertices. Since 1-trees are just trees, it is well-known that Gn,1 (after scaling the distances by 1/(2 √ n)) converges to the Continuum Random Tree Te. Our main result is that for k = 1, the random k-tree G n,k , scaled by (kH k−1 + 1)/(2 √ n) where H k−1 is the (k − 1)-th Harmonic number, converges to the Continuum Random Tree Te, too. In particular this shows that the diameter as well as the typical distance of two vertices in a random k-tree G n,k are of order √ n.