[Paper Review] Conditioning Galton-Watson trees on large maximal out-degree

Xin He|arXiv (Cornell University)|Dec 5, 2014

Stochastic processes and statistical mechanics5 references4 citations

TL;DR

This paper introduces a novel conditioning of Galton-Watson trees by their maximal out-degree, showing that critical trees conditioned on large maximal out-degree converge locally to Kesten’s tree (a size-biased tree with an infinite spine), while sub-critical trees converge to the condensation tree (a size-biased tree with a unique infinite node). The key contribution is establishing this conditioning as the most natural route to the condensation tree, completing the picture of tree limit behavior under extreme conditionings.

ABSTRACT

We propose a new way to condition random trees, that is, condition random trees to have large maximal out-degree. Under this new conditioning, we show that conditioned critical Galton-Watson trees converge locally to size-biased trees with a unique infinite spine. For the sub-critical case, we obtain local convergence to size-biased trees with a unique infinite node. We also study tail of the maximal out-degree of sub-critical Galton-Watson trees, which is essential for the proof of the local convergence.

Motivation & Objective

• To propose and analyze a new conditioning mechanism for Galton-Watson trees based on large maximal out-degree, which is more natural than previous conditionings like large height or large total progeny.
• To establish local convergence results for both critical and sub-critical Galton-Watson trees under this new conditioning.
• To complete the classification of limit behaviors of conditioned Galton-Watson trees by identifying maximal out-degree as the opposite extreme to height conditioning.
• To provide a new, elementary proof framework using the existing tools from Abraham and Delmas (2014), demonstrating the power of this new conditioning.
• To study the tail behavior of the maximal out-degree in sub-critical Galton-Watson trees, which is essential for proving convergence in the sub-critical case.

Proposed method

• Conditioning Galton-Watson trees on having a large maximal out-degree, defined as the maximum number of offspring across all vertices in the tree.
• Using the framework of local convergence from Abraham and Delmas (2014), which characterizes convergence to Kesten’s tree and condensation trees via size-biased exploration.
• Proving that for unbounded, critical offspring distributions, the conditioned tree converges locally to Kesten’s tree, which has a unique infinite spine.
• For unbounded, sub-critical offspring distributions, showing local convergence to the condensation tree, which features a unique infinite node with infinitely many subtrees.
• Establishing a key asymptotic equivalence between the tail of the offspring distribution and the tail of the maximal out-degree, up to a factor of $1 - \mu_p$, where $\mu_p$ is the mean offspring number.
• Leveraging continuity and convergence arguments in the space of rooted trees equipped with the local topology, using the distance $d_\infty$ from prior work.

Experimental results

Research questions

• RQ1 What happens to the local limit of a Galton-Watson tree when it is conditioned to have a large maximal out-degree?
• RQ2 Is conditioning on large maximal out-degree a more natural route to the condensation tree than conditioning on large total progeny or large height?
• RQ3 How does the tail behavior of the maximal out-degree relate to the tail of the offspring distribution in sub-critical Galton-Watson trees?
• RQ4 Can the local convergence to Kesten’s tree and the condensation tree be established under this new conditioning, and how do the proofs compare in simplicity to previous approaches?
• RQ5 What are the limitations of this conditioning for bounded offspring distributions or super-critical cases?

Key findings

• For unbounded, critical offspring distributions, the Galton-Watson tree conditioned on large maximal out-degree converges locally to Kesten’s tree, a size-biased tree with a unique infinite spine.
• For unbounded, sub-critical offspring distributions, the conditioned tree converges locally to the condensation tree, a size-biased tree with a unique infinite node.
• The tail of the maximal out-degree of a sub-critical Galton-Watson tree is asymptotically equivalent to the tail of the offspring distribution, up to a factor of $1 - \mu_p$, where $\mu_p$ is the mean offspring number.
• The proof technique is notably short and elementary, thanks to the convenient framework of Abraham and Delmas (2014), and the new conditioning simplifies the analysis by aligning naturally with the structure of the limit objects.
• For bounded offspring distributions, conditioning on large maximal out-degree is impossible for large $n$, so no local convergence to condensation or Kesten’s tree can occur, as shown in Remark 4.7.
• In the super-critical case, the limit is trivial: conditioning on non-extinction yields almost surely infinite maximal out-degree, and the distribution converges to the non-extinct version of the original tree, as shown in Remark 4.8.