Asymptotic height of Plancherel random trees
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arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866913069901283328 |
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| author | Zhang, Shengjun |
| author_facet | Zhang, Shengjun |
| contents | 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 $(T_n(θ))_{n \in \mathbb{N}}$ indexed by $θ>0$ such that $\frac{H_n}{\log n}$ converges in probability to $c_\star(θ)$, where $c_\star(θ)$ is an explicit constant depending on the parameter $θ$. The case of Plancherel trees corresponds to the parameter $θ=2$.
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 $c_\star(θ)$ is characterized through a variational principle associated with this branching random walk. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_25877 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Asymptotic height of Plancherel random trees Zhang, Shengjun Probability Combinatorics 60C05 (Primary), 05C80, 60J80, 05A16 (Secondary) 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 $(T_n(θ))_{n \in \mathbb{N}}$ indexed by $θ>0$ such that $\frac{H_n}{\log n}$ converges in probability to $c_\star(θ)$, where $c_\star(θ)$ is an explicit constant depending on the parameter $θ$. The case of Plancherel trees corresponds to the parameter $θ=2$. 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 $c_\star(θ)$ is characterized through a variational principle associated with this branching random walk. |
| title | Asymptotic height of Plancherel random trees |
| topic | Probability Combinatorics 60C05 (Primary), 05C80, 60J80, 05A16 (Secondary) |
| url | https://arxiv.org/abs/2604.25877 |