Kesten Tree in Random Tree Theory

• Kesten tree is a continuum random tree defined by an infinite spine with independently grafted subtrees via a Poisson point process.
• It displays self-similarity and infinite height, serving as a local limit for large, conditioned critical Galton–Watson trees.
• Its construction through scaling limits and Poisson representations highlights its role in connecting stable Lévy trees to discrete random tree models.

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 $\gamma \in (1,2]$ (Nassif, 2021). 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 $a^{1-1/\gamma} h$ (Nassif, 2021).

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 $\varepsilon \rightarrow 0$ (Nassif, 2021).

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 (Nassif, 2021).

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 (Nassif, 2021).

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 $S$ (Nassif, 2021).

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 (Nassif, 2021).