Percolation on grandparent graphs of Bienaymé trees

Abstract

We consider Bernoulli bond percolation on the grandparent graph of a Bienaymé tree, obtained by adding to the tree an edge between every vertex and its grandparent. We show that the critical parameter $p_c$ is the unique solution of an explicit equation involving the generating function of the offspring distribution, and we study how $p_c$ depends on this distribution beyond its mean $m$. When $m\ge 1.569$, the offspring distribution concentrated on the (one or two) integers closest to $m$ is the unique maximiser of $p_c$ among all offspring distributions with mean $m$, and its critical parameter is $\frac{1}{m^2}-\frac{1}{m^3}+O\left(\frac{1}{m^4}\right)$ as $m\to \infty$; more generally, for two distributions that are comparable in the convex order, $p_c$ is larger for the smaller of the two distributions under the condition that its support bounded some explicit $N(m)$ of order $m^4$. These restrictions cannot be removed: the convex order alone does not determine the order of the critical parameters, even among distributions with support bounded by $O(m^4)$, and the extremal property fails for $m$ close to $1$. Finally, for every $m>1$, the critical parameter $p_c$ can be arbitrarily close to $0$ for general offspring distributions with mean $m$. If we only consider offspring distributions that do not charge $0$, so that the obtained tree is almost surely infinite, this remains true if and only if $m\ge 2$.