Self-Similar Markov Tree (ssMt)

Updated 3 April 2026

Self-similar Markov trees are random, compact, rooted trees with a positive decoration function that scales in a self-similar manner.
They are constructed via recursive gluing of decorated segments using Lévy-driven processes and exhibit the Markov branching property for independent subtree evolution.
ssMt arise as scaling limits for various discrete trees, such as the Brownian CRT and stable Lévy trees, and provide genealogical codings in fragmentation and growth models.

A self-similar Markov tree (ssMt) is a random, compact, rooted real tree equipped with a positive decoration (labelling) function, engineered to satisfy both a self-similarity property and a Markov branching property. This structure generalizes a broad range of canonical tree-like random objects including the Brownian continuum random tree, Lévy trees, fragmentation trees, and growth-fragmentation trees. These trees arise as scaling limits of a variety of random discrete trees and serve as genealogical codings for systems in which branching, fragmentation, and growth dynamics follow self-similar Markovian rules, governed by Lévy processes and their time changes (Bertoin et al., 2024, Curien et al., 18 Dec 2025, Curien et al., 17 Mar 2026).

1. Characterization and Foundational Properties

A self-similar Markov tree is defined as a probability measure on the space of compact rooted real trees $(T, d, \rho)$ , endowed with an upper-semicontinuous decoration $g: T \to \mathbb{R}_+$ , required to be strictly positive on the skeletal part of the tree (Bertoin et al., 2024, Curien et al., 18 Dec 2025). The ssMt is characterized by two key properties:

Self-similarity: For each $x > 0$ , the law $\mathcal{Q}_x$ of a decorated tree with root label $g(\rho) = x$ satisfies

$\mathcal{Q}_x \text{(law of %%%%0%%%%)} = \mathcal{Q}_1 \text{(law of } (T, x^{\alpha} d, \rho, x g))$

for an index $\alpha > 0$ (Bertoin et al., 2024, Curien et al., 18 Dec 2025).

Markov-branching: For any base subtree $T'$ , the connected components of $T \setminus T'$ (with root values $(r_i, \ell_i)$ ) are, given the base, independent and distributed as $g: T \to \mathbb{R}_+$ 0.

The law of an ssMt is uniquely specified by a quadruple $g: T \to \mathbb{R}_+$ 1, where $g: T \to \mathbb{R}_+$ 2 is a drift, $g: T \to \mathbb{R}_+$ 3 a Gaussian coefficient, $g: T \to \mathbb{R}_+$ 4 a generalized Lévy measure on infinite sequences, and $g: T \to \mathbb{R}_+$ 5 the self-similarity parameter (Curien et al., 18 Dec 2025, Bertoin et al., 2024, Curien et al., 17 Mar 2026).

2. Genealogical and Growth-Fragmentation Construction

An ssMt is typically constructed as a recursive gluing of decorated line segments (with dynamics determined by a positive self-similar Markov process) following the Ulam–Harris tree structure (Bertoin et al., 2024, Curien et al., 18 Dec 2025). Specifically:

Each ancestor in the Ulam tree is associated with a trajectory given by a positive self-similar Markov process (pssMp) with the desired Lévy characteristics.
Placements for child gluing (branching points) are determined using a Poisson random measure on the timeline, with intensities coded by the Lévy measure $g: T \to \mathbb{R}_+$ 6.
The mass $g: T \to \mathbb{R}_+$ 7, or label, of each particle/branch evolves according to a Lamperti time-changed Lévy process:

$g: T \to \mathbb{R}_+$ 8

where $g: T \to \mathbb{R}_+$ 9 is the driving Lévy process determined by $x > 0$ 0, the Laplace exponent (Bertoin et al., 2024, Curien et al., 18 Dec 2025, Curien et al., 17 Mar 2026).

At random times of a Poisson process (with rate scaled by the current value of $x > 0$ 1), the branch fragments, splitting into several offspring with sizes determined via the Lévy measure and generalized to possibly multi-type settings (Curien et al., 18 Dec 2025, Stephenson, 2017). The recursive gluing under this regime produces a unique compact real tree structure satisfying the self-similarity and Markov branching axioms.

3. Scaling Limits and Discrete-to-Continuum Correspondence

Self-similar Markov trees serve as scaling limits for a wide variety of discrete random trees with consistent Markov branching properties. For suitable choices of splitting rules (encoded as splitting kernels or dislocation measures), and assuming regular variation for split-intensities, suitably rescaled finite trees converge in the Gromov–Hausdorff–Prokhorov topology to the corresponding ssMt (Haas et al., 2010, Pitman et al., 2012). Notable scaling limit results include:

Uniform unordered trees and critical Galton–Watson trees (after appropriate rescaling) converge to the Brownian continuum random tree (CRT), corresponding to the special case $x > 0$ 2 with the appropriate dislocation measure (Haas et al., 2010, Bertoin et al., 2024).
General Markov branching trees with rare macroscopic splits converge (when appropriately rescaled) to $x > 0$ 3-self-similar fragmentation trees with prescribed dislocation measures $x > 0$ 4 (Haas et al., 2010, Pitman et al., 2012).

The table below organizes several canonical models realized as ssMt scaling limits:

Model / Tree Type	$x > 0$ 5	Dislocation Measure / Lévy Data
Brownian CRT	$x > 0$ 6	Binary, $x > 0$ 7
Stable Lévy trees ( $x > 0$ 8)	$x > 0$ 9, $\mathcal{Q}_x$ 0	From ranked jumps of $\mathcal{Q}_x$ 1-stable subordinator
General fragmentations	$\mathcal{Q}_x$ 2	Arbitrary (subject to integrability and regular variation)

4. Explicit Measures, Spinal Decomposition, and Harmonic Analysis

Within a given ssMt, several explicit measures and decompositions arise:

Weighted length measures $\mathcal{Q}_x$ 3, where $\mathcal{Q}_x$ 4 is intrinsic length measure; $\mathcal{Q}_x$ 5 is finite iff the cumulant $\mathcal{Q}_x$ 6, with

$\mathcal{Q}_x$ 7

(Bertoin et al., 2024, Curien et al., 18 Dec 2025, Curien et al., 17 Mar 2026).

Harmonic (mass) measure: In the boundary case where the additive martingale degenerates (critical regime), a derivative martingale is constructed for harmonic measure via

$\mathcal{Q}_x$ 8

yielding a random measure $\mathcal{Q}_x$ 9 on the set of leaves, interpreted as canonical boundary mass (Curien et al., 17 Mar 2026).

Spinal decomposition: Conditioning the tree via length or harmonic measure leads to a decomposition wherein the path from root to a random typical point (the "spine") follows a tilted pssMp; all "dangling" subtrees attached along the spine are independent ssMt's, allowing for explicit recursive formulas (Bertoin et al., 2024, Curien et al., 18 Dec 2025, Curien et al., 17 Mar 2026).

5. Fractal Geometry and Dimensional Formulae

The Hausdorff dimension of the set of leaves of the tree $g(\rho) = x$ 0 is computable in terms of the index $g(\rho) = x$ 1 and the analytically determined minimizer $g(\rho) = x$ 2 of the cumulant: $g(\rho) = x$ 3 under critical (boundary) behavior ( $g(\rho) = x$ 4), or more generally as $g(\rho) = x$ 5 for multi-type or noncritical regimes ( $g(\rho) = x$ 6 the Malthusian exponent from a matrix equation when the underlying process is multitype) (Curien et al., 17 Mar 2026, Stephenson, 2017).

The exact distribution of root-to-leaf distances, height distributions, and other geometric functionals can often be described explicitly in terms of exponential functionals of the driving Lévy or Markov-additive process, and the corresponding moments can be computed by recursive or matrix-exponent formulas (Stephenson, 2017, Curien et al., 18 Dec 2025, Bertoin et al., 2024).

6. Critical Case and Martingale Techniques

Recent analysis of the critical case ( $g(\rho) = x$ 7) substantially extends the range of explicit results beyond the strict subcritical regime. Under differentiability and Cramér-type assumptions, critical ssMt can be constructed by careful control of compactness via branching random walk estimates (employing results such as (Aïdékon et al., 2024)), yielding sharp geometric and measure-theoretic descriptions, including limit theorems for convergence of length measures to the harmonic measure (Curien et al., 17 Mar 2026). In this context, the roles of additive and derivative martingales are crucial for constructing canonical measures and for explicit size-biasing of the law.

7. Self-Similarity in Random Trees Beyond Continuum Models

The ssMt formalism is equally applicable to the development of self-similar structure in discrete and combinatorial regimes. For example, for the class of level-set trees of symmetric homogeneous Markov chains (and their Brownian scaling limits), ssMt embodies both Horton and Tokunaga self-similarity laws, delivering explicit scaling ratios for occurrences of branch types and side-branching (Zaliapin et al., 2011). In this setting, the self-similar Markov tree provides a unified language to express universal branching statistics and side-branching patterns across both discrete and continuum models.

8. Canonical Examples and Generalizations

Brownian CRT: $g(\rho) = x$ 8, $g(\rho) = x$ 9, drift zero, and a binary splitting Lévy measure; the decoration can be interpreted as the mass of fringe subtrees or, in the excursion encoding, the local time (Bertoin et al., 2024, Curien et al., 18 Dec 2025).
Stable Lévy Trees: Indexed by $\mathcal{Q}_x \text{(law of %%%%0%%%%)} = \mathcal{Q}_1 \text{(law of } (T, x^{\alpha} d, \rho, x g))$0, $\mathcal{Q}_x \text{(law of %%%%0%%%%)} = \mathcal{Q}_1 \text{(law of } (T, x^{\alpha} d, \rho, x g))$1, with corresponding stable jump Lévy measure (Bertoin et al., 2024).
Multi-type and Extended Markov Trees: The ssMt structure is robust to generalization to multi-type fragmentations (jointly governed by Markov additive processes and vector-valued time-changes) and to extended self-similar fragmentations where fragmentation dynamics are coupled with evolving positive marks (Stephenson, 2017, Duchamps, 2019).

References

Bertoin, J.; Curien, N.; Riera, A. Self-similar Markov trees and scaling limits (Bertoin et al., 2024).
Curien, N.; Hu, Y.; Qian, J. Critical Self-Similar Markov Trees (Curien et al., 17 Mar 2026).
Haas, B.; Miermont, G. Scaling limits of Markov branching trees with applications to Galton–Watson and random unordered trees (Haas et al., 2010).
Pitman, J.; Rizzolo, D.; Winkel, M. Regenerative tree growth: structural results and convergence (Pitman et al., 2012).
Bertoin, J. Self-Similar Fragmentations (2006); Aldous, D. The Continuum Random Tree (1991); additional cited literature as listed within main texts.