Environment Scaling Methodology

• Environment scaling methodology is a framework that rigorously defines scaling limits for critical BGW trees with generation-dependent offspring distributions.
• It employs innovations such as martingale stopping, coarse-graining, and spine decomposition to overcome challenges from the loss of classical exchangeability.
• This approach proves that, under strict criticality and proper rescaling, large conditioned trees converge to the universal Brownian continuum random tree.

Environment scaling methodology, within the context of critical random trees in random environments, refers to the rigorous framework and analytical techniques for establishing scaling limits of families of Bienaymé–Galton–Watson (BGW) trees where the offspring law is randomly assigned to each generation. Specifically, each generation k is associated with its own offspring distribution μₖ, drawn i.i.d. from a common random law. In the "strictly critical" regime—where the mean of each μₖ equals 1 almost surely, and σₖ² = Var(ξₖ) is nondegenerate with finite expectation—this methodology demonstrates that large conditioned trees, under proper rescaling, converge to the Aldous Brownian continuum random tree (CRT). This necessitates new probabilistic and combinatorial tools, as classical results for constant-environment BGW trees do not hold due to the breakdown of exchangeability and Markov properties of traditional exploration processes (Conchon--Kerjan et al., 2022).

1. Strictly Critical BGW Trees in Random Environment

The environment scaling framework is situated within the paper of BGW trees where the generation-dependent offspring laws $\mu_k$ constitute a random environment, $(\mu_k)_{k\geq 0}$, i.i.d. across generations. The regime is "strictly critical": for all $k$,

$$\mathbb{E}[\xi_k] = 1 \quad\text{a.s.}, \qquad \sigma_k^2 = \operatorname{Var}(\xi_k)\in (0, \infty),\quad \mathbb{E}[\sigma_k^2]<\infty,$$

with $\xi_k \sim \mu_k$. This ensures almost sure finiteness of the BGW tree (no super- or subcriticality) while retaining sufficient variance for macroscopic fluctuations, key for identifying universal scaling limits.

2. Alternative Methodology for Scaling Limits

Classical scaling limit results for random trees (e.g., convergence to the CRT) rely on combinatorial exchangeability or Markovian properties of the Schensted or Lukasiewicz exploration processes, which no longer apply in a random environment due to non-identically distributed increments and lack of stationarity. To overcome these obstacles, the following innovations are central:

• Encoding by Lukasiewicz Path and Height Process: The tree is encoded by its Lukasiewicz path (encoding the succession of out-degrees minus one) and an associated height process. Whereas the former has i.i.d. increments in constant-environment models, in the random setting the increments’ law varies with generation, giving rise to non-identical distributions along the branch.
• Deterministic Regularity Conditions: Conditions (I)–(V) are imposed on the sequence $(\mu_k)_{k \geq 0}$, ensuring that average variances stabilize (condition (I)),

$$\frac{1}{n}\sum_{k=0}^{n-1}\sigma^2_k \to \sigma^2,$$

and that the frequency and impact of zero- or large-offspring events are controlled. These conditions are deterministic and verified almost surely for the random environment.

• Martingale Convergence and Stopping Techniques: Extending the martingale convergence theorem (as in Ethier–Kurtz and elaborated by Whitt), the proof works with stopped versions of the Lukasiewicz path, halting exploration upon escape from a rectangle of size growing as $\Theta(\sqrt{n})$, with $n$ the total progeny size. This preserves tightness and allows for Brownian approximation.
• Spine (Kesten Tree) Decomposition: Rather than relying on exchangeability, the analysis marks an infinite "spine" in a size-biased tree (the Kesten or Geiger tree). The exploration processes of the original (conditioned-large) and size-biased trees are compared, especially their height and reflected Lukasiewicz paths, whose properly normalized versions converge with the desired Brownian scaling.

3. Main Scaling Result

The chief result shows that BGW trees in an i.i.d. random environment, after conditioning to be large and rescaling,

$$\left(\mathcal{T}_n,\, \frac{1}{\sigma\sqrt{|\mathcal{T}_n|}} d_n,\, \pi_n\right) \xrightarrow{(d)} (\mathcal{T},d,\pi)$$

converge in the Gromov–Hausdorff–Prokhorov topology to the Brownian CRT, where $(\mathcal{T}_n, d_n, \pi_n)$ is the random tree with suitably normalized metric and probability measure, and $(\mathcal{T}, d, \pi)$ denotes Aldous' CRT. This extends the known limit for standard BGW trees to the random environment setting.

4. Technical Advances and Alternative Tools

• Blocking and Coarse-Graining: The tree is partitioned into genealogical "boxes" (micro-/mesoscopic control regions) via lexicographical and depth-first orders, providing control over local fluctuations and circumventing the lack of exchangeability.
• Stopped Exploration Processes: Growth-controlled rectangles $(h_n, w_n)$ define the scale at which stopping occurs—these sequences’ growth rate is $\Theta(\sqrt{n})$, allowing just enough flexibility for martingale approximation but not so large as to lose tightness.
• Refined Spine Arguments: The convergence of the ratio of the reflected Lukasiewicz path to the height process (to $\sigma^2/2$) along the spine is established, mirroring classical ratio limit theorems but in a non-exchangeable setting.
• Coarse Variance Averaging: Rather than law-of-large-numbers in identically distributed settings, deterministic block partitioning with careful averages (using the verified Conditions (I)–(V)) replace statistical stationarity to guarantee variance convergence at large scales.

5. Implications and Extended Relevance

• Universality of the Brownian CRT: The methodology demonstrates the robustness of Aldous' CRT as a universal scaling limit, even when offspring distributions are generation-dependent and randomly varying, as long as the system remains strictly critical with finite average variance.
• Broader Applications: These results are especially relevant for biological, ecological, and epidemic models where generational reproduction parameters (offspring variances) are environment-dependent or stochastic, such as seasonality or randomly fluctuating external conditions.
• New Analytical Framework: The developed techniques—especially the use of deterministic regularity conditions, martingale-stopping arguments, and the spine decomposition—offer a blueprint for analyzing more general classes of trees and branching structures beyond the reach of classical probabilistic tools.

6. Comparison to Constant Environment and Future Directions

• Loss of Exchangeability and Markov Properties: The approach fundamentally differs from standard BGW settings where tools like Donsker's invariance principle exploit i.i.d. increments. Here, block decomposition and spine-based reasoning are essential substitutes.
• Extensions and Open Problems: The methodology sets the stage for further investigations into more general environments (beyond i.i.d. across generations), near-critical or heavy-tailed offspring regimes, and even explicit analysis of genealogical structure or metric measure spaces in random environmental contexts.

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In summary, environment scaling methodology for critical BGW trees in a random environment provides a rigorous, tool-rich approach to proving that conditioned large trees converge to the Brownian continuum random tree, despite the loss of classical probabilistic simplifications. This is achieved through a union of deterministic averaging, martingale analysis, careful stopping strategies, and innovative spine decompositions, collectively extending scaling limit theory to a significantly broader and more realistic class of random tree models (Conchon--Kerjan et al., 2022).