Growing conditioned BGW trees with log-concave offspring distributions

TL;DR

This paper demonstrates that conditioned BGW trees with log-concave offspring distributions can be realized as a Markov process adding leaves, with applications to stochastic ordering in inhomogeneous percolation models.

Contribution

It introduces a Markov process construction for conditioned BGW trees with log-concave offspring distributions and extends previous results to new classes of distributions.

Findings

Markov process construction for conditioned BGW trees with log-concave offspring distributions

Extension to offspring distributions supported on arithmetic progressions

Existence of increasing couplings in inhomogeneous Bernoulli percolation models

Abstract

We show that given a log-concave offspring distribution, the corresponding sequence of Bienaym\'e-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.