Self-Similar Markov Trees (ssMt)

• Self-similar Markov trees are random, decorated real trees featuring scaling invariance and strong Markov branching, capturing the continuum limits of discrete branching models.
• They are constructed via growth-fragmentation processes, SDE couplings, and the Lamperti transformation, which together ensure a coherent framework for scaling limits.
• Applications include analyzing Brownian and stable CRTs, multi-type fragmentation trees, and determining volume growth and fractal properties through martingale techniques.

A self-similar Markov tree (ssMt) is a random compact real tree equipped with a decoration function, typically interpreted as "mass," "size," or "height" along branches, such that the tree exhibits both self-similarity under scaling and satisfies a strong Markov branching property. The class of ssMt encompasses key objects in random geometry and probability, including continuum random trees (CRT), stable trees, fragmentation and growth-fragmentation trees, and, in multi-type generalization, trees with marked genealogies or types. The construction and scaling limits of ssMt unify diverse models, particularly as limits of discrete branching structures such as multi-type Markov branching trees, Galton-Watson processes, and regenerative tree growth. The ssMt framework has enabled precise analysis of scaling exponents, limit shapes, volume growth, and the dimensional structure of random trees, and establishes natural invariance principles for discrete-to-continuum limits in probabilistic combinatorics and stochastic processes.

1. Decorated Real Trees, Self-Similarity, and Markov Branching

An ssMt is formally a decorated real tree $\mathtt{T} = (T, d_T, \rho, g)$, where $(T,d_T)$ is a compact real tree rooted at $\rho$, and $g:T\to[0,\infty)$ is an upper-semicontinuous decoration function, usually encoding "mass" or "height" that is strictly positive along the skeleton and jumps down at branch points (Bertoin et al., 2024). Self-similarity of index $\alpha>0$ manifests as follows: under $\mathbb{Q}_1$, the law of the rescaled tree $(T, x^\alpha d_T, \rho, xg)$ is again $\mathbb{Q}_x$ for every $x>0$. The Markov branching property requires that for any height $h>0$, conditionally on the decorated truncation $B_h(T)$, the connected components of $T \setminus B_h(T)$ are independent and distributed as $\mathbb{Q}_y$, where $y$ is the decoration at their root.

Crucially, the construction of ssMt unifies deterministic gluing of line segments (indexed by genealogical structures such as the Ulam tree) and randomness introduced via Lévy processes and growth-fragmentation mechanisms. This framework generalizes classical combinatorial tree models and embeds the Lamperti transformation for positive self-similar Markov processes (pssMp) as the foundational building block (Bertoin et al., 2024, Curien et al., 18 Dec 2025).

2. Construction of ssMt via Growth-Fragmentation and Scaling Limits

The law of an ssMt is determined by a quadruple $(a, \sigma^2, \Lambda; \alpha)$ specifying drift $a$, diffusion $\sigma^2$, a splitting measure $\Lambda$ on the infinite simplex (encoding dislocation laws for masses upon splitting), and a self-similarity index $\alpha$ (Curien et al., 18 Dec 2025). Particles ("fragments") grow and split according to their current label $x$, which evolves as a solution to a Markovian SDE with drift $a x^{1-\alpha}$, diffusion $\sigma^2 x^{2-\alpha}$, and jump-driven splitting at rate $x^{-\alpha}\Xi(\mathrm{d}y)$ for a dislocation measure $\Xi$.

The tree is recursively constructed by gluing sample paths of the pssMp along each branch, where the Lamperti transformation ties together the Lévy process coding a fragment's evolution and the global genealogical tree structure (Bertoin et al., 2024, Stephenson, 2017). Branching points are determined by jump times, with sizes of offspring given by the image of the jump sizes under a time-change, and the overall tree is built by recursively grafting independent subtrees at each split.

Scaling limits of discrete Markov branching trees, subject to "macroscopic rarity" conditions, converge in the Gromov–Hausdorff–Prokhorov (GHP) topology to ssMt (Haas et al., 2010, Haas et al., 2019). The key regime is when the rescaled first-split distributions converge to a dislocation measure, with the self-similarity index determined by regular variation exponents in the discrete model.

3. Multi-Type Extensions, Markov Additive Processes, and Scaling Regimes

In the multi-type setting, individuals have both size and type $(n, i)$, and splitting kernels $q_n^{(i)}$ govern the joint distribution of children's sizes and types (Haas et al., 2019). The scaling limit yields multi-type self-similar fragmentation trees, for which the associated process along a tagged path is a Markov additive process (MAP): $(\xi_t, J_t)$ with $\xi_t$ a type-dependent subordinator and $J_t$ a finite-state Markov chain determining type (Stephenson, 2017). The dislocation measures $\bar\nu^{(i)}$ then encode both mass and type upon splitting.

Three fundamental scaling regimes arise based on the relative rates of type change versus macroscopic splitting:

• Critical regime: type changes are of the same order as large splits, yielding genuine multi-type fragmentation trees.
• Solo regime: type change is negligible; limits are monotype fragmentation trees indexed by the initial type.
• Mixing regime: type mixing dominates prior to splitting; the limit is a monotype tree with a stationary type-averaged dislocation measure (Haas et al., 2019).

The convergence in metric measure topology (often GHP) characterizes the limiting trees, their measures, and scaling exponents—e.g., Brownian CRT for critical Galton-Watson with finite variance, stable CRT for heavy tail offspring (Pagnard, 2016).

4. Dislocation Measures, Martingales, and Fractal Geometry

Dislocation measures $\nu$ (or $\bar\nu^{(i)}$ in the multi-type case) encode the distribution over fragment sizes produced by splitting and are central objects in both discrete and continuum ssMt (Haas et al., 2010, Haas et al., 2019). Scaling limits require the convergence of the rescaled discrete splitting measures to dislocation measures in the space of decreasing sequences (or annotated with types).

Self-similar Markov trees are naturally equipped with weighted length and harmonic mass measures, both constructed via martingale arguments and spinal decompositions (Bertoin et al., 2024). Additive martingales, constructed from the sum of mass-specific functionals over branches, yield critical exponents which determine the Hausdorff dimension of the leaves. Under Malthusian hypotheses, the dimension is given by $p^*/|\alpha|$, where $p^*$ solves the associated eigenvalue problem for the Laplace exponent of the process (Stephenson, 2017).

Spinal decompositions further provide a powerful recursive framework: biasing by total mass (length or harmonic) selects a spine along which the process is tilted, and off-spine subtrees are independent ssMt with root size determined by the branching law (Bertoin et al., 2024).

5. Applications and Special Examples

ssMt encompasses and generalizes several fundamental families:

• Brownian CRT: characterized by $\alpha=1/2$, binary conservative splitting, and mass measure corresponding to the contour measure; arises as scaling limit of uniform and critical GW trees with finite variance (Pagnard, 2016, Curien et al., 18 Dec 2025).
• Stable CRT: for heavy-tailed offspring laws, $\alpha=1-1/\beta$ with $\beta \in (1,2]$, corresponding splitting measure, and explicit scaling exponent (Haas et al., 2010, Bertoin et al., 2024).
• Growth-fragmentation trees: blend mass growth and splitting, with explicit SDE characterizations (Curien et al., 18 Dec 2025).
• Multi-type fragmentation and the genealogy of multi-type Markov branching trees: application to conditioned Galton-Watson models and leaf-growth algorithms (Haas et al., 2019).
• Generalization to random decorated trees with arbitrary type structure and growth rules: new invariance principles and coupling constructions (Curien et al., 18 Dec 2025, Bertoin et al., 2024).

Volume growth exponents for the limiting trees ($\beta=-1/\alpha$) and scaling relations are derived from the structure of the underlying fragmentation-immigration Poisson processes, providing explicit asymptotics for mass within metric balls (Pagnard, 2016).

6. Methodologies, Invariance, and Open Problems

The convergence from discrete models to ssMt relies on establishing tightness in the GHP topology, martingale bounds, and identification of finite-dimensional marginals via marked path analysis and induction (Haas et al., 2010, Haas et al., 2019). The key is functional limit theorems for the residual mass process, the embedding of Markov chains as subordinators, and the construction of ssMt as scaling limits under regular variation and convergence of dislocation measures (Pitman et al., 2012).

Recent work leverages explicit SDE couplings (G-growing framework) to build nested monotone families of ssMt, generalizing Aldous's line-breaking construction and providing a new viewpoint on the scaling limits of leaf-growth algorithms (Curien et al., 18 Dec 2025).

Throughout, open questions include the characterization of ssMt under criticality ($\min \kappa =0$), the fractal structure of harmonic measures, connections to random planar maps (the cactus, peeling trees), and the universality of the ssMt paradigm for scaling limits of random geometric and combinatorial structures (Bertoin et al., 2024).