The Critical Beta-splitting Random Tree IV: Mellin analysis of Leaf Height

In the critical beta-splitting model of a random $n$-leaf rooted tree, clades are recursively split into sub-clades, and a clade of $m$ leaves is split into sub-clades containing $i$ and $m-i$ leaves with probabilities $\propto 1/(i(m-i))$. The height of a uniform random leaf can be represented as the absorption time of a certain {\em harmonic descent} Markov chain. Recent work on these heights $D_n$ and $L_n$ (corresponding to discrete or continuous versions of the tree) has led to quite sharp expressions for their asymptotic distributions, based on their Markov chain description. This article gives even sharper expressions, based on an $n \to \infty$ limit tree structure described via exchangeable random partitions in the style of Haas et al (2008). Within this structure, calculations of moments lead to expressions for Mellin transforms, and then via Mellin inversion we obtain sharp estimates for the expectation, variance, Normal approximation and large deviation behavior of $D_n$.