Asymptotics for Beta-Splitting Trees via Homogeneous Fragmentations and Meromorphic Potential Theory

Abstract

Inspired by recent work of Aldous, Janson, and Pittel on the critical beta-splitting model, we study the full beta-splitting family for beta greater than minus two through a canonical continuous-time embedding into a homogeneous exchangeable fragmentation. In this representation, the frequency of a tagged fragment is described by a subordinator. We express the continuous height of a typical leaf, its occupation probabilities, the discrete height, and the total continuous-time length in terms of the potential measure of this subordinator. Renewal theory yields first-order asymptotics and a central limit theorem for the continuous height. A regenerative-composition representation gives Gaussian limits for the discrete height above and at the critical value, and a non-Gaussian power-law limit below it. We also obtain residue expansions for the potential measure and the mean continuous height using meromorphic potential theory and generalized Nevanlinna functions. Finally, we study the maximum continuous-time height, proving a law of large numbers and a mixed Gumbel limit. At the critical parameter value, this resolves an open problem of Aldous and Janson.