Growth-Fragmentation Trees Overview

• Growth-fragmentation trees are continuum random trees that encode the genealogical structure of systems where particles grow and split according to self-similar dynamics.
• They are constructed via self-similar growth-fragmentation processes from Lévy processes and serve as universal scaling limits for discrete random trees.
• Intrinsic martingales and boundary measures in these trees facilitate statistical inference, linking probabilistic behavior with analytical growth-fragmentation equations.

A growth-fragmentation tree is a continuum random tree whose genealogical structure encodes the evolution of a branching system where particles (or cells, fragments, clusters) grow according to a prescribed dynamics and split into offspring according to specified fragmentation rules, often with inheritable or Markovian structure. Growth-fragmentation trees appear as universal scaling limits of discrete growing and fragmenting trees (such as random recursive trees or $k$-ary trees), arise as the genealogical trees of self-similar growth-fragmentation processes constructed from Lévy processes, and underlie the geometry of random structures in probability, statistical physics, and biology.

1. Genealogical and Analytical Structure of Growth-Fragmentation Trees

Growth-fragmentation trees formalize the genealogy of a growth-fragmentation process: a branching Markov system in which each "cell" or "particle" evolves according to a self-similar Markov process (often a positive self-similar Markov process, pssMp) and, at random (often jump) times, splits into multiple offspring, which then evolve independently. The genealogical structure is captured by a (measured) random real tree, often a continuum random tree (CRT), equipped with a mass or area measure on its boundary corresponding to the populations of descendants, and a consistent scaling/self-similarity property (Ged, 2017, Bertoin et al., 2016, Pitman et al., 2013, Gall et al., 2018). The process is fully characterized by its self-similarity index $\alpha<0$ and a dislocation (or "fragmentation") measure $\nu$ on ranked mass partitions.

Formally, the tree $(T, d, \rho, \mu)$ is a compact metric space with root $\rho$ and Borel probability measure $\mu$ supported on its leaves, wherein for each "height" $t>0$, the subtrees above level $t$ form independent copies of $T$ rescaled in metric by mass-powers $m_i^\gamma$ and in measure by $\alpha<0$0, with $\alpha<0$1 typically related to $\alpha<0$2 by $\alpha<0$3 (Pitman et al., 2012, Haas et al., 2014). The genealogy of a tagged cell is described by a self-similar Markov process derived from the underlying mass fragmentation structure.

2. Construction via Self-Similar Growth-Fragmentation

Growth-fragmentation trees can be rigorously constructed from self-similar growth-fragmentation processes built from Lévy processes with only negative jumps (or, more generally, suitable Markov additive processes). Given a spectrally negative Lévy process $\alpha<0$4, the associated positive self-similar process is defined via Lamperti time-change: $\alpha<0$5 with index $\alpha<0$6 (Bertoin et al., 2016, Gall et al., 2018, Ged, 2017).

Each negative jump of $\alpha<0$7 constitutes a fragmentation event, giving birth to a new daughter of corresponding mass while the parent continues. The full growth-fragmentation process $\alpha<0$8 is the decreasingly ordered vector of present active fragment masses. The process is fully characterized by its cumulant (or Laplace) exponent $\alpha<0$9, which combines the Lévy exponent and fragmentation law: $\nu$0 A key property is the existence of a Malthusian parameter $\nu$1 (minimal positive root $\nu$2), which dictates scaling and the existence of additive (martingale) structures. The intrinsic area measure of the tree, supported on its boundary, is then canonically defined as the martingale limit of total $\nu$3-th power masses along rays (Bertoin et al., 2016).

3. Scaling Limits and Universality

A rich class of discrete random trees have scaling limits described by growth-fragmentation trees. For instance, $\nu$4-ary recursive trees grown by random edge-splitting and attachment of $\nu$5 leaves scale under appropriate normalization ($\nu$6 scaling of edge lengths at size $\nu$7) to self-similar fragmentation trees of index $\nu$8 (Haas et al., 2014). The CRT of Aldous arises as the scaling limit in the $\nu$9 case ($(T, d, \rho, \mu)$0); for $(T, d, \rho, \mu)$1, the limit is a genuine fragmentation tree differing from stable Lévy trees. Embedding structures allow nested limiting trees as $(T, d, \rho, \mu)$2 varies.

Random recursive trees fragmented by node removals, or in general more sophisticated random graph evolution models, exhibit size distributions and scaling limits governed by power law tails whose exponents are explicitly computed from the process parameters and universal across a range of models (Kalay et al., 2014, Bansaye et al., 2021). The rate-equation approach provides tractable formulas for fragment-size densities and their temporal evolution.

Self-similar growth-fragmentations also arise as scaling limits of statistics in random planar maps and statistical physics models. The perimeter process in Markovian explorations of large Boltzmann random maps converges to a distinguished family of growth-fragmentation processes indexed by a stability parameter, further identifying the law of the CRT measure as a size-biased stable law (Bertoin et al., 2016, Watson, 2021, Ged, 2017).

4. Characterization via Martingales, Measures, and Profiles

A central role in the analysis of growth-fragmentation trees is played by intrinsic additive martingales,

$(T, d, \rho, \mu)$3

which, under minimal assumptions, converge in $(T, d, \rho, \mu)$4 and almost surely to a finite random limiting variable—the intrinsic area of the tree (Bertoin et al., 2016). The corresponding boundary measure $(T, d, \rho, \mu)$5 on the leaves is constructed as the projective limit of mass measures on successively finer cylinders indexed by genealogical rays, providing a full measure-geometric structure for the genealogical tree (Ged, 2017).

The regularity of the profile, specifically whether $(T, d, \rho, \mu)$6 is absolutely continuous or singular with respect to Lebesgue measure, is governed by the comparison $(T, d, \rho, \mu)$7 (Ged, 2017). When this holds, the density $(T, d, \rho, \mu)$8 of the cumulative area profile can be computed from the small-mass asymptotics of fragments, while in the singular regime the Hausdorff dimension of the measure is strictly less than 1 and given explicitly by $(T, d, \rho, \mu)$9.

5. Specific Models: Brownian CRTs, Ricocheted Stable Processes, and Markov Branchings

Special cases yield explicit and highly tractable growth-fragmentation trees. The Brownian CRT corresponds to a growth-fragmentation tree with $\rho$0 and splitting density $\rho$1, and its bead splitting construction coincides with Aldous’s line-breaking construction (Pitman et al., 2013, Gall et al., 2018). General bead-splitting processes provide a universal construction for $\rho$2-self-similar CRTs by iterating random grafting of "strings of beads" encoding the instantaneous fragmenter subordinator.

The model of Brownian motion indexed by the Brownian tree leads to a canonical growth-fragmentation tree when slicing the tree at height $\rho$3 and considering the sequence of boundary sizes of connected components; this process has an explicit Lamperti representation and dislocation measure supported on $\rho$4 (Gall et al., 2018). Spines of growth-fragmentation trees are probabilistically described by conditioned (ricocheted) stable processes, which illuminate the scaling and tail behavior of fragment sizes and connect to notions in conformal loop ensembles and Liouville quantum gravity (Watson, 2021).

Markov branching models with regenerative structure—where the splitting of subtrees at branch points is independent, as encoded by a dislocation measure—admit precise representation theorems and necessary and sufficient conditions for scaling limits to self-similar fragmentation trees, expressed in Gromov–Hausdorff–Prokhorov topology (Pitman et al., 2012). General bead-splitting, Ulam–Harris labeled cell systems, and paths to tagged leaves furnish canonical couplings between process and tree measure structures.

6. Analytical and Statistical Aspects

Growth-fragmentation trees admit statistical inference frameworks grounded in empirical observation of genealogical trees of growing cell populations. Piecewise deterministic Markov branching trees with size-dependent division rates and Markovian growth rates yield mean-field PDEs—growth-fragmentation equations—whose solutions describe the expected empirical measure over size, growth rate, and, in extensions, additional marks such as bacterial pole asymmetry (Doumic et al., 2012, Krell, 2024). Nonparametric estimators for division rates have been constructed and shown to achieve optimal minimax rates under mild regularity conditions by leveraging the tree-observational structure.

The many-to-one formula links the empirical moments over the entire growing tree to those of a single tagged cell whose evolution is governed by the same generator, yielding a duality between the probabilistic structure of the tree and the analytic structure of the growth-fragmentation PDE.

7. Applications and Extensions

Growth-fragmentation trees provide fundamental models for universal limit objects arising in random compliant trees, random planar maps, statistical mechanics of disordered media, biological branching and cell-division mechanisms, and the geometry of scaling limits in high-dimensional random structures (Ged, 2017, Gall et al., 2018, Bertoin et al., 2016, Watson, 2021). Recent applications include quantification of epidemic contact-tracing and cluster sizes in random recursive trees subject to dynamic isolation and fragmentation (Bansaye et al., 2021), and structural analysis of fragmentation phenomena in random networks and evolving forests (Kalay et al., 2014).

Extensions include non-exchangeable models (alpha-theta/gamma), bead-splitting processes beyond binary fragmentation, and incorporation of additional state variables or controlled interventions in the growth and fragmentation dynamics. The universality and robustness of growth-fragmentation trees continue to motivate further research into scaling limits, genealogical measures, and the interplay with combinatorial and probabilistic representations of complex systems.