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

TL;DR

This paper analyzes a generalized Derrida-Retaux model on a Galton-Watson tree with geometric offspring, characterizing the critical curve and confirming the Derrida-Retaux conjecture for certain initial distributions.

Contribution

It introduces a generalized model on a geometric Galton-Watson tree, characterizes the critical curve via an involution equation, and proves the free energy satisfies the conjecture.

Findings

Critical curve characterized by an involution equation

Free energy satisfies the Derrida-Retaux conjecture

Applicable to geometric and exponential initial distributions

Abstract

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.