Growing conditioned BGW trees with log-concave offspring distributions

TL;DR

This work establishes that BGW trees conditioned on size with log-concave offspring laws admit monotone increasing couplings when grown as a Markov process that adds a right-leaning leaf at each step. It extends the construction to offspring distributions supported on arithmetic progressions by adding right-leaning bouquets of leaves and proves analogous results for simply generated trees, connecting to random compositions through admissibility and Toeplitz TP2 properties. The authors derive elementary, self-contained proofs and apply the framework to an inhomogeneous model of random subtrees on the Ulam–Harris tree, which corresponds to an increasing root-cluster size in an inhomogeneous Bernoulli percolation model. This generalizes Luczak and Winkler’s uniform-subtree couplings and clarifies the role of log-concavity in enabling stochastic monotonicity, while also outlining open questions regarding the necessity of log-concavity beyond certain supports.

Abstract

We show that given a log-concave offspring distribution, the corresponding sequence of Bienaymé-Galton-Watson trees conditioned to have $n\geq 1$ vertices admits a realization as a Markov process $(T_n)_{n\geq1}$ which adds a new "right-leaning" leaf at each step. This applies for instance to offspring distributions which are Poisson, binomial, geometric, or any convolution of those. By a negative result of Janson, the log-concavity condition is optimal in the restricted case of offspring distributions supported in $\{0,1,2\}$. We then prove a generalization to the case of an offspring distribution supported on an arithmetic progression, if we assume log-concavity along that progression. As an application, we deduce the existence of increasing couplings in an inhomogeneous model of random subtrees of the Ulam--Harris tree. This is equivalent to the statement that, in a corresponding inhomogeneous Bernouilli percolation model on a regular tree, the root cluster is stochastically increasing in its size. These results generalize a construction of Luczak and Winkler which applies to uniformly sampled subtrees with $n$ vertices of the infinite complete $d$-ary trees. Our proofs are elementary and we tried to make them as self-contained as possible.

Figures (11)

• Figure 1: An increasing sequence of rooted plane trees as obtained in Theorem \ref{['thm:main-thm-BGW']}. The root is at the bottom and the children of a vertex are ordered left-to-right. At each step, a new right-leaning leaf is added.
• Figure 2: An increasing sequence of rooted subtrees of the infinite complete $d$-ary tree, as in Luczak and Winkler's coupling, here with $d=3$.
• Figure 3: An illustration of the action of the mappings $\mathsf{push}$ and $\mathsf{comp}^{(d)}$, here with $d=3$. Notice that $\mathsf{comp}^{(d)}$ preserves the order $\subseteq$ while $\mathsf{push}$ does not.
• Figure 4: An example of an increasing sequence of rooted plane trees, as in Theorem \ref{['thm:main-thm-BGW']}. Observe that the new leaves at each step (in yellow) must be right-leaning. For convenience, we only represented a portion of the Ulam--Harris tree, namely the bottom part of its subset $\mathbb{U}^{(3)}$.
• Figure 5: An example of an increasing sequence of rooted plane trees as obtained in Theorem \ref{['thm:main-thm-BGW-arithmetic']} with $d=3$. Observe that at each step, a new right-leaning bouquet of $d$ leaves is added (in yellow). For convenience, we only represented a portion of the Ulam--Harris tree, namely the bottom part of its subset $\mathbb{U}^{(6)}$.

Theorems & Definitions (119)

• Theorem 1
• Theorem 2
• Theorem 3
• Remark 1.1: Percolation interpretation
• Remark 2.1
• Remark 2.2
• Definition 2.3: Admissibility
• Lemma 2.4
• proof
• Lemma 2.5: Key lemma