---
title: Kesten Tree in Random Tree Theory
url: https://www.emergentmind.com/topics/kesten-tree
type: topic
---

# Kesten Tree in Random Tree Theory

The Kesten tree is a continuum random tree characterized by a distinguished infinite spine along which independent subtrees are grafted according to a Poisson point process. This object emerges as the scaling limit when zooming in at the root of a normalized stable Lévy tree and serves as the local limit for large conditioned critical Galton–Watson trees with offspring distributions in the domain of attraction of a stable law of index $\gamma \in (1,2]$ [2103.13649]. The Kesten tree possesses self-similarity, infinite height, and a branching structure governed by the same mechanism as the original stable tree.

## 1. Stable Lévy Trees and Their Scaling

Let $\gamma \in (1,2]$. The $\gamma$-stable Lévy tree, defined under its excursion measure $\mathcal{N}$, is a random compact rooted real tree $(T,d,\mu)$ with total mass $\sigma = \mu(T)$ and height $h = \sup_{x\in T} d(\rho,x)$. Their distributions satisfy
\[
\mathcal{N}[\sigma \in da] = (\gamma \Gamma(1-1/\gamma))^{-1} a^{-1-1/\gamma}\,da,\quad \mathcal{N}[h \in da] = (γ-1)^{-γ/(γ-1)} a^{-γ/(γ-1)} da.
\]
The tree exhibits self-similarity: for any $a>0$, $R_\gamma((T,d,\mu),a) := (T, a\cdot d, a^{\gamma/(\gamma-1)}\mu)$. The normalized stable tree is $\mathbb{P}^1 := \mathcal{N}[\cdot \mid \sigma=1]$. If $(T,d,\mu)\sim\mathbb{P}^1$, then scaling as
\[
\mathcal{T}^{(a)} := R_\gamma(T, a^{1-1/\gamma})
\]
produces a tree of mass $a$ and height $a^{1-1/\gamma} h$ [2103.13649].

## 2. Construction of the Unnormalized Kesten Tree

Zooming in at the root of a normalized stable tree at speed $\varepsilon \rightarrow 0$ yields a random marked tree converging in distribution to the unnormalized Kesten tree $\mathcal{K}$. For $(T,d,\mu)\sim\mathbb{P}^1$, choosing a random μ-leaf $U \in T$ and examining the unique branch $[\rho,U]$, the grafted subtrees $\{T_i, i \in I\}$ have heights $h_i$ and masses $\sigma_i$. Define
\[
\mathcal{N}_{\varepsilon} = \sum_{h_i \leq \varepsilon H(U)} \delta_{(\varepsilon^{-1} h_i,\; \varepsilon^{-\gamma/(\gamma-1)} \sigma_i,\; T_i)}.
\]
As $\varepsilon \to 0$ and with rescaling $f(\varepsilon) = \varepsilon$, the random marked tree $(T, H(U), \mathcal{N}_{\varepsilon})$ converges in the Gromov–Hausdorff–Prokhorov sense to $(\mathcal{K}, H, \sum_{s\geq 0}\delta_{(s,T'_s)})$, where $(T'_s, s \geq 0)$ is a Poisson point process. Scaling distances by $\varepsilon^{-1}$ and masses by $\varepsilon^{-\gamma/(\gamma-1)}$, $R_\gamma(T, \varepsilon^{-1}) \xrightarrow{d} \mathcal{K}$ as $\varepsilon \rightarrow 0$ [2103.13649].

## 3. Poisson Point Process Representation

The Kesten tree $\mathcal{K}$ is represented as follows:
- Start with an infinite spine (the half-line $[0,\infty)$, rooted at $0$).
- Graft subtrees at times $s \ge 0$ along the spine, determined by a Poisson point process of intensity $ds \otimes \Pi(dT)$, where the measure $\Pi$ on rooted compact trees is
  \[
  \Pi(dT) = 
  \begin{cases}
      2\mathcal{N}(dT) & \text{if } \gamma=2,\\
      \int_0^\infty r\,\pi(dr)\,\mathbb{P}^{(r)}(dT) & \text{if } 1<\gamma<2,
  \end{cases}
  \]
  with $\pi(dr) = \gamma(\gamma-1)\Gamma(2-\gamma)^{-1} r^{-1-\gamma} dr$ and $\mathbb{P}^{(r)}$ the law of the forest started from mass $r$.
- The resulting metric space has spine length measure (Lebesgue on $[0,\infty)$) and on each grafted subtree its intrinsic mass–measure [2103.13649].

## 4. Main Scaling and Convergence Theorems

Let $(T,d,\mu)\sim\mathbb{P}^1$ be a normalized stable tree of index $\gamma$ and $U$ a $\mu$-leaf. Define, for $\varepsilon > 0$,
\[
T^{(\varepsilon)} = R_\gamma(T, \varepsilon^{-1}),
\]
and the point measure
\[
\mathcal{N}_{\varepsilon} = \sum_{h_i \leq \varepsilon H(U)} \delta_{(\varepsilon^{-1}h_i, \varepsilon^{-\gamma/(\gamma-1)}\sigma_i, T_i)}.
\]
As $\varepsilon \to 0$, for any Lipschitz test function $\Phi$,
\[
\langle\mathcal{N}_{\varepsilon}, \Phi\rangle \to \sum_{s \ge 0} \Phi(s, \mu(T'_s), T'_s)
\]
in distribution. Consequently,
\[
(T^{(\varepsilon)}, \mu) \xrightarrow[\varepsilon\to 0]{d} \mathcal{K}
\]
in Gromov–Hausdorff–Prokhorov topology. In discrete terms, for normalized $\gamma$-stable trees $(\mathcal{T}_n, d_n, \mu_n)$ at mass-level $n$ and $n^{1/\gamma}d_n$ rescaling,
\[
(\mathcal{T}_n, n^{1/\gamma}d_n, \mu_n) \xrightarrow[n\to\infty]{d} \mathcal{K} [2103.13649].
\]

## 5. Properties of the Kesten Tree

- **Branching Mechanism:** The local branching follows the same process with $\psi(\lambda) = \lambda^\gamma$ as the underlying stable tree.
- **Infinite Spine:** There exists a unique geodesic ray from the root with infinite length.
- **Mass-Measure:** The total mass is infinite; the measure $\mu$ is $\sigma$-finite, with linear density along the spine determined by the Poisson intensity.
- **Height Distribution:** The height to infinity along the spine is infinite; grafted finite trees obey the height law of $\mathcal{N}$.
- **Self-Similarity:** For any $a>0$, $R_\gamma(\mathcal{K}, a) \overset{d}{=} a\mathcal{K}$, so $\mathcal{K}$ is self-similar with index 1 [2103.13649].

## 6. Additive Functionals and Asymptotic Limits

On a compact real tree $(T,d,\mu)$, additive functionals of the form
\[
Z_{\alpha, \beta}(T) = \int_T \mu(dx) \int_0^{H(x)} \sigma_{r,x}^\alpha h_{r,x}^\beta\,dr
\]
are of interest, where $\sigma_{r,x}$ is the mass and $h_{r,x}$ the height of the subtree above level $r$ containing $x$. For the normalized tree as $\max(\alpha,\beta)\to\infty$:
- **Subcritical regime:** If $\beta/\alpha^{1-1/\gamma} \rightarrow c \in [0,\infty)$,
  \[
  \alpha^{1-1/\gamma} h^{-\beta} Z_{\alpha, \beta}(T) \xrightarrow{d} \int_0^\infty e^{-S_t - c t/h} dt,
  \]
  where $(S_t)$ is a stable subordinator with Laplace exponent $\gamma\lambda^{1-1/\gamma}$ and $h=\operatorname{height}(T)$.
- **Supercritical regime:** If $\beta/\alpha^{1-1/\gamma}\rightarrow\infty$,
  \[
  \beta h^{-\beta} Z_{\alpha,\beta}(T) \xrightarrow{\mathbb{P}} h.
  \]
The limits correspond to integrals along the infinite spine of the Kesten tree with dynamics governed by a subordinator $S$ [2103.13649].

## 7. Connections with Critical Galton–Watson Trees

Critical Galton–Watson (GW) trees whose offspring distribution lies in the domain of attraction of a stable law of index $\gamma$ and are conditioned to have $n$ vertices, converge after rescaling edge lengths by $n^{-1/\gamma}$ to the normalized stable tree. The *local* limit of such GW trees seen from the root is the discrete Kesten tree. In the continuum, this yields the Kesten tree $\mathcal{K}$, which emerges as the scaling limit and natural local-limit object for large conditioned GW trees. The Poisson-grafting decomposition of the Kesten tree mirrors the decomposition of a critical GW tree into subcritical forests grafted along its infinite spine [2103.13649].

Source: https://www.emergentmind.com/topics/kesten-tree