Galton-Watson trees with first ancestor interaction

Abstract: We consider the set of random Bienaym\'e-Galton-Watson trees with a bounded number of offspring and bounded number of generations as a statistical mechanics model: a random tree is a rooted subtree of the maximal tree; the spin at a given node of the maximal tree is equal to the number of offspring if the node is present in the random tree and equal to -1 otherwise. We introduce nearest neighbour interactions favouring pairs of neighbours which both have a relatively large offspring. We then prove (1) correlation inequalities and (2) recursion relations for generating functions, mean number of external nodes, interaction energy and the corresponding variances. The resulting quadratic dynamical system, in two dimensions or more depending on the desired number of moments, yields almost exact numerical results. The balance between offspring distribution and coupling constant leads to a phase diagram for the analogue of the extinction probability. On the transition line the mean number of external nodes in generation $n+1$ is found numerically to scale as $n^{-2}$.