Asymptotic height of Plancherel random trees

Abstract

We study a natural analogue of Ulam's problem for random rooted trees distributed according to a Plancherel-type measure. This probability measure is closely related to the classical Plancherel measure on integer partitions. For a Plancherel random tree $T_n$ with $n$ vertices, we investigate the asymptotic behavior of its height $H_n$, defined as the maximal distance from the root to a leaf. We prove that this height grows logarithmically. More precisely, there is a one-parameter family of random trees $(T_n(\theta))_{n \in \mathbb{N}}$ indexed by $\theta>0$ such that $\frac{H_n}{\log n}$ converges in probability to $c_\star(\theta)$, where $c_\star(\theta)$ is an explicit constant depending on the parameter $\theta$. The case of Plancherel trees corresponds to the parameter $\theta=2$. The proof is based on the fact that the Plancherel random trees can be viewed as Ewens fragmentation trees, for which the height exhibits a sharp threshold phenomenon. An upper bound is obtained via $s$-mass functionals and contraction estimates, while the lower bound is derived by embedding the model into a branching random walk with logarithmic displacements governed by a Poisson--Dirichlet distribution. The constant $c_\star(\theta)$ is characterized through a variational principle associated with this branching random walk.