Year: 2025
Type: article
Abstract: 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 ∝1∕(i(m−i)). The height of a uniform random leaf can be represented as the absorption time of a certain harmonic descent Markov chain. Recent work on these heights Ln and Dn (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→∞ 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 Dn.
DOI: 10.1214/25-ejp1332
Authors: David Aldous, Svante Janson
Institutions: University of California, Berkeley, Uppsala University
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: gold
Estimated APC paid: $675
OpenAlex ID: W4409460525