The Derrida-Retaux model on a geometric Galton-Watson tree

We consider a generalized Derrida-Retaux model on a Galton-Watson tree with a geometric offspring distribution. For a class of recursive systems, including the Derrida-Retaux model with either a geometric or exponential initial distribution, we characterize the critical curve using an involution-type equation and prove that the free energy satisfies the Derrida-Retaux conjecture.